K11a359

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K11a358.gif

K11a358

K11a360.gif

K11a360

Contents

K11a359.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X8291 X14,4,15,3 X16,6,17,5 X18,8,19,7 X22,10,1,9 X20,12,21,11 X2,14,3,13 X6,16,7,15 X4,18,5,17 X12,20,13,19 X10,22,11,21
Gauss code 1, -7, 2, -9, 3, -8, 4, -1, 5, -11, 6, -10, 7, -2, 8, -3, 9, -4, 10, -6, 11, -5
Dowker-Thistlethwaite code 8 14 16 18 22 20 2 6 4 12 10
A Braid Representative
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A Morse Link Presentation K11a359 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a359/ThurstonBennequinNumber
Hyperbolic Volume 9.67188
A-Polynomial See Data:K11a359/A-polynomial

[edit Notes for K11a359's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a359's four dimensional invariants]

Polynomial invariants

Alexander polynomial 6 t^2-13 t+15-13 t^{-1} +6 t^{-2}
Conway polynomial 6 z^4+11 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, 4 }
Jones polynomial -q^{13}+q^{12}-3 q^{11}+5 q^{10}-6 q^9+8 q^8-8 q^7+8 q^6-6 q^5+4 q^4-2 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +2 z^4 a^{-6} +2 z^4 a^{-8} +z^4 a^{-10} +2 z^2 a^{-4} +4 z^2 a^{-6} +4 z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} + a^{-6} + a^{-8} + a^{-10} -2 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +3 z^9 a^{-11} +z^9 a^{-13} +3 z^8 a^{-8} -3 z^8 a^{-10} -5 z^8 a^{-12} +z^8 a^{-14} +3 z^7 a^{-7} -6 z^7 a^{-9} -13 z^7 a^{-11} -3 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -9 z^6 a^{-8} +6 z^6 a^{-10} +14 z^6 a^{-12} -4 z^6 a^{-14} +2 z^5 a^{-5} -6 z^5 a^{-7} +9 z^5 a^{-9} +23 z^5 a^{-11} -6 z^5 a^{-15} +z^4 a^{-4} -6 z^4 a^{-6} +14 z^4 a^{-8} -7 z^4 a^{-10} -25 z^4 a^{-12} +3 z^4 a^{-14} -3 z^3 a^{-5} +6 z^3 a^{-7} -4 z^3 a^{-9} -21 z^3 a^{-11} +3 z^3 a^{-13} +11 z^3 a^{-15} -2 z^2 a^{-4} +5 z^2 a^{-6} -6 z^2 a^{-8} +2 z^2 a^{-10} +16 z^2 a^{-12} +z^2 a^{-14} +5 z a^{-11} -z a^{-13} -6 z a^{-15} - a^{-6} + a^{-8} - a^{-10} -2 a^{-12}
The A2 invariant Data:K11a359/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a359/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (11, 36)
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
44 288 968 \frac{7450}{3} \frac{1286}{3} 12672 23392 4160 3584 \frac{42592}{3} 41472 \frac{327800}{3} \frac{56584}{3} \frac{6676661}{30} -\frac{3294}{5} \frac{4380202}{45} \frac{30955}{18} \frac{405461}{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=4 is the signature of K11a359. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25            0
23         31 -2
21        2   2
19       43   -1
17      42    2
15     44     0
13    44      0
11   24       2
9  24        -2
7  2         2
512          -1
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=9 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=10 {\mathbb Z}
r=11 {\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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K11a358.gif

K11a358

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K11a360