K11a124

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K11a123.gif

K11a123

K11a125.gif

K11a125

Contents

K11a124.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a124 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X16,12,17,11 X8,14,9,13 X22,16,1,15 X12,18,13,17 X6,20,7,19 X18,22,19,21
Gauss code 1, -5, 2, -1, 3, -10, 4, -7, 5, -2, 6, -9, 7, -3, 8, -6, 9, -11, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 10 14 20 2 16 8 22 12 6 18
A Braid Representative
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A Morse Link Presentation K11a124 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a124/ThurstonBennequinNumber
Hyperbolic Volume 17.524
A-Polynomial See Data:K11a124/A-polynomial

[edit Notes for K11a124's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a124's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^3-18 t^2+34 t-41+34 t^{-1} -18 t^{-2} +5 t^{-3}
Conway polynomial 5 z^6+12 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 155, 6 }
Jones polynomial -q^{14}+5 q^{13}-11 q^{12}+17 q^{11}-23 q^{10}+25 q^9-25 q^8+21 q^7-14 q^6+9 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +11 z^4 a^{-8} -z^4 a^{-10} -z^4 a^{-12} +3 z^2 a^{-6} +13 z^2 a^{-8} -9 z^2 a^{-10} + a^{-6} +5 a^{-8} -7 a^{-10} +2 a^{-12}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +13 z^9 a^{-11} +8 z^9 a^{-13} +6 z^8 a^{-8} +14 z^8 a^{-10} +21 z^8 a^{-12} +13 z^8 a^{-14} +3 z^7 a^{-7} -10 z^7 a^{-11} +4 z^7 a^{-13} +11 z^7 a^{-15} +z^6 a^{-6} -14 z^6 a^{-8} -38 z^6 a^{-10} -45 z^6 a^{-12} -17 z^6 a^{-14} +5 z^6 a^{-16} -6 z^5 a^{-7} -20 z^5 a^{-9} -25 z^5 a^{-11} -27 z^5 a^{-13} -15 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +16 z^4 a^{-8} +34 z^4 a^{-10} +23 z^4 a^{-12} +4 z^4 a^{-14} -4 z^4 a^{-16} +3 z^3 a^{-7} +25 z^3 a^{-9} +34 z^3 a^{-11} +17 z^3 a^{-13} +5 z^3 a^{-15} +3 z^2 a^{-6} -13 z^2 a^{-8} -19 z^2 a^{-10} -3 z^2 a^{-12} -10 z a^{-9} -13 z a^{-11} -3 z a^{-13} - a^{-6} +5 a^{-8} +7 a^{-10} +2 a^{-12}
The A2 invariant Data:K11a124/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a124/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, 16)
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
28 128 392 \frac{2306}{3} \frac{286}{3} 3584 \frac{15584}{3} \frac{2528}{3} 512 \frac{10976}{3} 8192 \frac{64568}{3} \frac{8008}{3} \frac{1096537}{30} \frac{40606}{15} \frac{483314}{45} \frac{3143}{18} \frac{37177}{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=6 is the signature of K11a124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          4 4
25         71 -6
23        104  6
21       137   -6
19      1210    2
17     1313     0
15    812      -4
13   613       7
11  38        -5
9  6         6
713          -2
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=6 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=7 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=8 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=9 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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