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

Visit K11a51 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X14,5,15,6 X10,8,11,7 X2,9,3,10 X20,11,21,12 X16,13,17,14 X6,15,7,16 X22,18,1,17 X12,19,13,20 X18,22,19,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -10, 7, -3, 8, -7, 9, -11, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 8 14 10 2 20 16 6 22 12 18
A Braid Representative
A Morse Link Presentation K11a51 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a51/ThurstonBennequinNumber
Hyperbolic Volume 15.1159
A-Polynomial See Data:K11a51/A-polynomial

[edit Notes for K11a51'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 K11a51's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-9 t^2+28 t-39+28 t^{-1} -9 t^{-2} + t^{-3}
Conway polynomial z^6-3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 115, -2 }
Jones polynomial q^3-3 q^2+7 q-12+16 q^{-1} -18 q^{-2} +19 q^{-3} -16 q^{-4} +12 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-4 z^2 a^4-2 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+3 a^2-2 z^4-3 z^2-2+z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+6 a^3 z^9+3 a z^9+5 a^6 z^8+9 a^4 z^8+8 a^2 z^8+4 z^8+5 a^7 z^7+7 a^5 z^7+a^3 z^7+2 a z^7+3 z^7 a^{-1} +3 a^8 z^6-2 a^6 z^6-11 a^4 z^6-13 a^2 z^6+z^6 a^{-2} -6 z^6+a^9 z^5-7 a^7 z^5-20 a^5 z^5-20 a^3 z^5-16 a z^5-8 z^5 a^{-1} -5 a^8 z^4-6 a^6 z^4-6 a^4 z^4-5 a^2 z^4-3 z^4 a^{-2} -3 z^4-2 a^9 z^3+4 a^7 z^3+16 a^5 z^3+16 a^3 z^3+13 a z^3+7 z^3 a^{-1} +3 a^8 z^2+7 a^6 z^2+10 a^4 z^2+10 a^2 z^2+3 z^2 a^{-2} +7 z^2+a^9 z-a^7 z-4 a^5 z-4 a^3 z-4 a z-2 z a^{-1} -a^8-2 a^6-2 a^4-3 a^2- a^{-2} -2
The A2 invariant Data:K11a51/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a51/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a3, K11a331,}

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{350}{3} \frac{82}{3} -128 -\frac{1472}{3} -\frac{128}{3} -160 \frac{32}{3} 512 \frac{1400}{3} \frac{328}{3} \frac{66511}{30} -\frac{7502}{15} \frac{67502}{45} \frac{2033}{18} \frac{6991}{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 K11a51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          2 -2
3         51 4
1        72  -5
-1       95   4
-3      108    -2
-5     98     1
-7    710      3
-9   59       -4
-11  27        5
-13 15         -4
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

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