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

Visit K11a130 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X20,7,21,8 X16,9,17,10 X2,11,3,12 X18,13,19,14 X22,16,1,15 X12,17,13,18 X8,19,9,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -10, 5, -2, 6, -9, 7, -3, 8, -5, 9, -7, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 10 14 20 16 2 18 22 12 8 6
A Braid Representative
A Morse Link Presentation K11a130 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:K11a130/ThurstonBennequinNumber
Hyperbolic Volume 15.9906
A-Polynomial See Data:K11a130/A-polynomial

[edit Notes for K11a130'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 K11a130's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-12 t^2+29 t-37+29 t^{-1} -12 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 123, -2 }
Jones polynomial -q^2+4 q-8+14 q^{-1} -17 q^{-2} +20 q^{-3} -20 q^{-4} +16 q^{-5} -12 q^{-6} +7 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-2 z^4 a^6-3 z^2 a^6-a^6+z^6 a^4+z^4 a^4-z^2 a^4-2 a^4+z^6 a^2+2 z^4 a^2+3 z^2 a^2+3 a^2-z^4-z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-8 z^5 a^9+5 z^3 a^9-z a^9+5 z^8 a^8-13 z^6 a^8+11 z^4 a^8-6 z^2 a^8+a^8+5 z^9 a^7-11 z^7 a^7+10 z^5 a^7-8 z^3 a^7+3 z a^7+2 z^{10} a^6+6 z^8 a^6-26 z^6 a^6+32 z^4 a^6-15 z^2 a^6+a^6+11 z^9 a^5-27 z^7 a^5+31 z^5 a^5-20 z^3 a^5+8 z a^5+2 z^{10} a^4+9 z^8 a^4-27 z^6 a^4+26 z^4 a^4-6 z^2 a^4-2 a^4+6 z^9 a^3-6 z^7 a^3+2 z^5 a^3-4 z^3 a^3+4 z a^3+8 z^8 a^2-11 z^6 a^2+2 z^4 a^2+3 z^2 a^2-3 a^2+7 z^7 a-10 z^5 a+2 z^3 a+4 z^6-6 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-1}
The A2 invariant Data:K11a130/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a130/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a78,}

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

Vassiliev invariants

V2 and V3: (-1, 4)
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 32 8 -\frac{254}{3} \frac{14}{3} -128 -\frac{160}{3} -\frac{256}{3} -32 -\frac{32}{3} 512 \frac{1016}{3} -\frac{56}{3} \frac{43409}{30} \frac{1102}{15} \frac{34258}{45} \frac{367}{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 K11a130. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          3 3
1         51 -4
-1        93  6
-3       96   -3
-5      118    3
-7     99     0
-9    711      -4
-11   59       4
-13  27        -5
-15 15         4
-17 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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