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

Visit K11a136 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,6,17,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X8,13,9,14 X20,15,21,16 X22,18,1,17 X12,19,13,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -7, 5, -2, 6, -10, 7, -4, 8, -3, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 16 14 18 2 8 20 22 12 6
A Braid Representative
A Morse Link Presentation K11a136 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a136/ThurstonBennequinNumber
Hyperbolic Volume 18.0711
A-Polynomial See Data:K11a136/A-polynomial

[edit Notes for K11a136'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 K11a136's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-14 t^2+39 t-53+39 t^{-1} -14 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 163, -2 }
Jones polynomial -q^2+5 q-11+18 q^{-1} -23 q^{-2} +27 q^{-3} -26 q^{-4} +22 q^{-5} -16 q^{-6} +9 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8-2 z^4 a^6-z^2 a^6+z^6 a^4-z^2 a^4-a^4+z^6 a^2+z^4 a^2+2 z^2 a^2+2 a^2-z^4
Kauffman polynomial (db, data sources) z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-9 z^5 a^9+6 z^3 a^9-z a^9+7 z^8 a^8-14 z^6 a^8+8 z^4 a^8-2 z^2 a^8+7 z^9 a^7-8 z^7 a^7-5 z^5 a^7+8 z^3 a^7-3 z a^7+3 z^{10} a^6+12 z^8 a^6-37 z^6 a^6+28 z^4 a^6-6 z^2 a^6+17 z^9 a^5-28 z^7 a^5+7 z^5 a^5+6 z^3 a^5-2 z a^5+3 z^{10} a^4+19 z^8 a^4-47 z^6 a^4+28 z^4 a^4-z^2 a^4-a^4+10 z^9 a^3-5 z^7 a^3-12 z^5 a^3+7 z^3 a^3+14 z^8 a^2-20 z^6 a^2+6 z^4 a^2+2 z^2 a^2-2 a^2+11 z^7 a-14 z^5 a+3 z^3 a+5 z^6-4 z^4+z^5 a^{-1}
The A2 invariant Data:K11a136/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a136/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: (1, -1)
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 -8 8 \frac{110}{3} \frac{58}{3} -32 -\frac{464}{3} -\frac{224}{3} -40 \frac{32}{3} 32 \frac{440}{3} \frac{232}{3} \frac{15871}{30} \frac{458}{15} \frac{20822}{45} -\frac{1087}{18} \frac{1951}{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 K11a136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          4 4
1         71 -6
-1        114  7
-3       138   -5
-5      1410    4
-7     1213     1
-9    1014      -4
-11   612       6
-13  310        -7
-15 16         5
-17 3          -3
-191           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\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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