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(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a323 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,5,17,6 X22,8,1,7 X20,10,21,9 X4,12,5,11 X18,13,19,14 X10,15,11,16 X2,17,3,18 X14,19,15,20 X8,22,9,21
Gauss code 1, -9, 2, -6, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -7, 10, -5, 11, -4
Dowker-Thistlethwaite code 6 12 16 22 20 4 18 10 2 14 8
A Braid Representative
A Morse Link Presentation K11a323 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a323/ThurstonBennequinNumber
Hyperbolic Volume 13.4473
A-Polynomial See Data:K11a323/A-polynomial

[edit Notes for K11a323's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11a323's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-9 t^2+19 t-23+19 t^{-1} -9 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 83, -2 }
Jones polynomial q^5-3 q^4+5 q^3-8 q^2+11 q-12+13 q^{-1} -11 q^{-2} +9 q^{-3} -6 q^{-4} +3 q^{-5} - q^{-6}
HOMFLY-PT polynomial (db, data sources) a^2 z^6+z^6-a^4 z^4+3 a^2 z^4-2 z^4 a^{-2} +3 z^4-2 a^4 z^2+3 a^2 z^2-5 z^2 a^{-2} +z^2 a^{-4} +4 z^2-a^4+a^2-3 a^{-2} + a^{-4} +3
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+6 a z^9+9 z^9 a^{-1} +3 z^9 a^{-3} +9 a^2 z^8-5 z^8 a^{-2} +z^8 a^{-4} +3 z^8+10 a^3 z^7-14 a z^7-40 z^7 a^{-1} -16 z^7 a^{-3} +9 a^4 z^6-21 a^2 z^6-9 z^6 a^{-2} -5 z^6 a^{-4} -34 z^6+6 a^5 z^5-19 a^3 z^5+52 z^5 a^{-1} +27 z^5 a^{-3} +3 a^6 z^4-13 a^4 z^4+8 a^2 z^4+29 z^4 a^{-2} +8 z^4 a^{-4} +45 z^4+a^7 z^3-4 a^5 z^3+6 a^3 z^3+6 a z^3-20 z^3 a^{-1} -15 z^3 a^{-3} +5 a^4 z^2-18 z^2 a^{-2} -5 z^2 a^{-4} -18 z^2+a^5 z-a z+z a^{-1} +z a^{-3} -a^4-a^2+3 a^{-2} + a^{-4} +3
The A2 invariant Data:K11a323/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a323/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_83, K11a307,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (1, -2)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
4 -16 8 -\frac{34}{3} -\frac{62}{3} -64 -\frac{448}{3} \frac{320}{3} -176 \frac{32}{3} 128 -\frac{136}{3} -\frac{248}{3} \frac{5071}{30} \frac{646}{5} -\frac{2338}{45} -\frac{1135}{18} -\frac{2129}{30}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=-2 is the signature of K11a323. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           11
9          2 -2
7         31 2
5        52  -3
3       63   3
1      65    -1
-1     76     1
-3    57      2
-5   46       -2
-7  25        3
-9 14         -3
-11 2          2
-131           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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