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

Visit K11a64 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X20,11,21,12 X22,13,1,14 X18,15,19,16 X6,17,7,18 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 8 16 10 2 20 22 18 6 14 12
A Braid Representative
A Morse Link Presentation K11a64 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a64/ThurstonBennequinNumber
Hyperbolic Volume 14.0131
A-Polynomial See Data:K11a64/A-polynomial

[edit Notes for K11a64's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a64's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-22 t+27-22 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 97, -4 }
Jones polynomial 1-2 q^{-1} +5 q^{-2} -9 q^{-3} +13 q^{-4} -15 q^{-5} +16 q^{-6} -14 q^{-7} +11 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-a^{10}+2 z^4 a^8+3 z^2 a^8-z^6 a^6-z^4 a^6+3 z^2 a^6+3 a^6-z^6 a^4-3 z^4 a^4-4 z^2 a^4-3 a^4+z^4 a^2+3 z^2 a^2+2 a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-5 z^4 a^{12}+2 z^2 a^{12}+5 z^7 a^{11}-8 z^5 a^{11}+4 z^3 a^{11}-z a^{11}+5 z^8 a^{10}-6 z^6 a^{10}+3 z^4 a^{10}-3 z^2 a^{10}+a^{10}+3 z^9 a^9+z^7 a^9-5 z^5 a^9+2 z^3 a^9+z^{10} a^8+5 z^8 a^8-7 z^6 a^8+2 z^4 a^8+5 z^9 a^7-7 z^7 a^7+7 z^5 a^7-7 z^3 a^7+2 z a^7+z^{10} a^6+2 z^8 a^6-11 z^4 a^6+11 z^2 a^6-3 a^6+2 z^9 a^5-z^7 a^5-3 z^5 a^5+2 z^3 a^5-z a^5+2 z^8 a^4-z^6 a^4-9 z^4 a^4+11 z^2 a^4-3 a^4+2 z^7 a^3-6 z^5 a^3+5 z^3 a^3-z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 a^2
The A2 invariant Data:K11a64/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a64/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n174,}

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

Vassiliev invariants

V2 and V3: (4, -11)
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
16 -88 128 \frac{1448}{3} \frac{208}{3} -1408 -\frac{8944}{3} -\frac{1408}{3} -440 \frac{2048}{3} 3872 \frac{23168}{3} \frac{3328}{3} \frac{285542}{15} \frac{112}{15} \frac{347648}{45} \frac{2602}{9} \frac{14822}{15}

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 K11a64. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1           11
-1          1 -1
-3         41 3
-5        62  -4
-7       73   4
-9      86    -2
-11     87     1
-13    68      2
-15   58       -3
-17  26        4
-19 15         -4
-21 2          2
-231           -1
Integral Khovanov Homology

(db, data source)

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