Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links

By Louis H. Kauffman, D. Radford. 2001

133 citations recorded by the source corpus.

This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. We then introduce the concept of a bi-oriented quantum algebra, which provides an algebraic context for this structure. In this paper we discuss the construction of a generalized Alexander polynomial GK(s;t) via the concept of a biquandle. Our approach leads directly to a gener- alization of the Burau representation upon which this invariant is based. We then reformulate the invariant as a quantum link invariant and as a state summation. In this context we show that the normalized quantum invariant ZK(ae;? ) satisfles a Conway skein identity and reproduces GK(s;t). These invariants are useful for the theory of virtual knots and links, for they vanish on classical knots and links. Hence the invariants studied here can be used to show that many virtual knots and links are not classical. We give such examples in Section 2. We also give an example of a virtual knot that cannot be detected by the generalized Alexander polynomial. This knot is detected by the structure of the corresponding generalized Alexander module. We conclude with a diagram due to Kishino that is knotted, but is not detected by any representation of the biquandle at the time of this writing. We conjecture that the biquandle of the Kishino diagram does detect its knottedness. In the flnal section of the paper, we formulate the concept of a bi-oriented quantum algebra. This generalizes our previous concept of oriented quantum alge- bra (6, 7) to include the necessary structures to create invariants of virtual links. The invariant ZK(ae;? ) studied in this paper flts non-trivially into this framework. Subsequent papers will study the structure and applications of bi-oriented quantum algebras.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e1c188aa50e6e8154cc1ea6849767fd5bca795d4

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Citations in the corpus

Virtual knots and links (2005, background): “In some cases one has an invariant of virtuals that is an extension of ideas from classical knot theory that vanishes or is otherwise trivial for classical knots. See [Saw],[SW], [ KR ], [Ma2, Ma3, Ma5].”
VIRTUAL BRAIDS AND THE L-MOVE (2005, unknown): “For reference to previous work on virtual links and braids the reader should consult [2, 6, 7, 9, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22 , 23, 27, 28, 32, 36, 37].”
Virtual knot groups and almost classical knots (2015, methodology): “In [3], we proved a formula relating the virtual Alexander polynomial HK(s, t, q) to the generalized Alexander polynomial GK(s, t) defined by Sawollek [30], and in equivalent forms by Kauffman–Radford [19], Manturov [23], and Silver–Williams [31, 32].”
Virtual knot theory-unsolved problems (2005, unknown): “…[4] [5] [9] [10] [11] [12] [8] [13] [16] [17] [18] [19] [20] [22] [23] [24] [25] [26] [31] [33] [34] [36] [37] [38] [39] [40] [41] [43] [50] [51] [55] [58] [59] [60] [61] [62] [64] [65] [67] [68] [70] [72] [73] [74] [75] [76] [78] [79] [80] [81] [82] [84] [86] [87] [88] [83] [90] [93] [94] [95]…”
Matrices and finite biquandles (2006, background): “M uch recentwork hasbeen doneon theknotinvariantsde ned by counting hom om orphism sfrom a knotquandleintoa nitetargetquandle([9],[3],etc.).”