K11a123

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K11a122.gif

K11a122

K11a124.gif

K11a124

Contents

K11a123.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X8,12,9,11 X18,14,19,13 X22,16,1,15 X6,18,7,17 X12,20,13,19 X16,22,17,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -10, 7, -3, 8, -11, 9, -7, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 10 14 20 2 8 18 22 6 12 16
A Braid Representative
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A Morse Link Presentation K11a123 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a123/ThurstonBennequinNumber
Hyperbolic Volume 15.9713
A-Polynomial See Data:K11a123/A-polynomial

[edit Notes for K11a123'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 K11a123's four dimensional invariants]

Polynomial invariants

Alexander polynomial 9 t^2-29 t+41-29 t^{-1} +9 t^{-2}
Conway polynomial 9 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 117, 4 }
Jones polynomial -q^{13}+4 q^{12}-8 q^{11}+12 q^{10}-17 q^9+18 q^8-18 q^7+17 q^6-11 q^5+7 q^4-3 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +3 z^4 a^{-6} +4 z^4 a^{-8} +z^4 a^{-10} +z^2 a^{-4} +4 z^2 a^{-6} +6 z^2 a^{-8} -3 z^2 a^{-10} -z^2 a^{-12} +2 a^{-6} +2 a^{-8} -4 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +6 z^9 a^{-9} +11 z^9 a^{-11} +5 z^9 a^{-13} +9 z^8 a^{-8} +11 z^8 a^{-10} +6 z^8 a^{-12} +4 z^8 a^{-14} +8 z^7 a^{-7} -2 z^7 a^{-9} -25 z^7 a^{-11} -14 z^7 a^{-13} +z^7 a^{-15} +6 z^6 a^{-6} -14 z^6 a^{-8} -40 z^6 a^{-10} -34 z^6 a^{-12} -14 z^6 a^{-14} +3 z^5 a^{-5} -9 z^5 a^{-7} -20 z^5 a^{-9} +z^5 a^{-11} +6 z^5 a^{-13} -3 z^5 a^{-15} +z^4 a^{-4} -7 z^4 a^{-6} +8 z^4 a^{-8} +33 z^4 a^{-10} +31 z^4 a^{-12} +14 z^4 a^{-14} -2 z^3 a^{-5} +4 z^3 a^{-7} +23 z^3 a^{-9} +20 z^3 a^{-11} +6 z^3 a^{-13} +3 z^3 a^{-15} -z^2 a^{-4} +6 z^2 a^{-6} -2 z^2 a^{-8} -12 z^2 a^{-10} -6 z^2 a^{-12} -3 z^2 a^{-14} -9 z a^{-9} -12 z a^{-11} -4 z a^{-13} -z a^{-15} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12}
The A2 invariant Data:K11a123/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a123/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: (7, 17)
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
28 136 392 \frac{2594}{3} \frac{358}{3} 3808 \frac{18352}{3} \frac{3040}{3} 712 \frac{10976}{3} 9248 \frac{72632}{3} \frac{10024}{3} \frac{1339417}{30} \frac{33766}{15} \frac{678794}{45} \frac{5447}{18} \frac{55897}{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 K11a123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          3 3
23         51 -4
21        73  4
19       105   -5
17      87    1
15     1010     0
13    78      -1
11   410       6
9  37        -4
7  4         4
513          -2
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}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=7 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a122

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K11a124