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

Visit K11a54 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X14,6,15,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X10,19,11,20 X6,21,7,22
Gauss code 1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -3, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 14 16 2 18 20 22 12 10 6
A Braid Representative
A Morse Link Presentation K11a54 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:K11a54/ThurstonBennequinNumber
Hyperbolic Volume 16.7029
A-Polynomial See Data:K11a54/A-polynomial

[edit Notes for K11a54'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 K11a54's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-13 t^2+33 t-43+33 t^{-1} -13 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 139, -2 }
Jones polynomial -q^2+5 q-10+16 q^{-1} -20 q^{-2} +23 q^{-3} -22 q^{-4} +18 q^{-5} -13 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^6-2 a^6+z^6 a^4+z^4 a^4+z^2 a^4+a^4+z^6 a^2+z^4 a^2-z^4+1
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+6 z^3 a^9-2 z a^9+5 z^8 a^8-12 z^6 a^8+10 z^4 a^8-5 z^2 a^8+a^8+5 z^9 a^7-8 z^7 a^7+2 z^5 a^7+3 z^3 a^7-2 z a^7+2 z^{10} a^6+9 z^8 a^6-31 z^6 a^6+34 z^4 a^6-14 z^2 a^6+2 a^6+12 z^9 a^5-21 z^7 a^5+11 z^5 a^5-z^3 a^5+z a^5+2 z^{10} a^4+15 z^8 a^4-37 z^6 a^4+28 z^4 a^4-8 z^2 a^4+a^4+7 z^9 a^3-14 z^5 a^3+5 z^3 a^3+z a^3+11 z^8 a^2-14 z^6 a^2+2 z^4 a^2-z^2 a^2+10 z^7 a-14 z^5 a+3 z^3 a+5 z^6-5 z^4+1+z^5 a^{-1}
The A2 invariant Data:K11a54/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a54/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a172,}

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

Vassiliev invariants

V2 and V3: (-1, 3)
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 24 8 -\frac{158}{3} \frac{38}{3} -96 -80 -64 -72 -\frac{32}{3} 288 \frac{632}{3} -\frac{152}{3} \frac{29969}{30} -\frac{526}{5} \frac{32698}{45} \frac{1039}{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 K11a54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          4 4
1         61 -5
-1        104  6
-3       117   -4
-5      129    3
-7     1011     1
-9    812      -4
-11   510       5
-13  28        -6
-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}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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