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

Visit K11a56 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X16,6,17,5 X2837 X18,9,19,10 X20,11,21,12 X22,13,1,14 X6,16,7,15 X14,17,15,18 X12,19,13,20 X10,21,11,22
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -9, 8, -3, 9, -5, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 8 16 2 18 20 22 6 14 12 10
A Braid Representative
A Morse Link Presentation K11a56 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a56/ThurstonBennequinNumber
Hyperbolic Volume 14.5406
A-Polynomial See Data:K11a56/A-polynomial

[edit Notes for K11a56's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a56's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-25 t+33-25 t^{-1} +11 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 { 109, 0 }
Jones polynomial -q^5+3 q^4-7 q^3+12 q^2-15 q+18-17 q^{-1} +15 q^{-2} -11 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6-z^6+a^4 z^4-3 a^2 z^4+2 z^4 a^{-2} -z^4+2 a^4 z^2-5 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +2 z^2+a^4-3 a^2+ a^{-2} - a^{-4} +3
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +4 a^4 z^8+7 a^2 z^8+5 z^8 a^{-2} +8 z^8+3 a^5 z^7-a^3 z^7-4 a z^7+5 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-9 a^4 z^6-17 a^2 z^6-3 z^6 a^{-2} +3 z^6 a^{-4} -13 z^6-9 a^5 z^5-10 a^3 z^5-8 a z^5-15 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+5 a^4 z^4+13 a^2 z^4-4 z^4 a^{-2} -5 z^4 a^{-4} +6 z^4+8 a^5 z^3+12 a^3 z^3+10 a z^3+11 z^3 a^{-1} +3 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-a^4 z^2-8 a^2 z^2+4 z^2 a^{-2} +3 z^2 a^{-4} -4 z^2-2 a^5 z-5 a^3 z-5 a z-3 z a^{-1} +z a^{-5} +a^4+3 a^2- a^{-2} - a^{-4} +3
The A2 invariant Data:K11a56/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a56/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a185, K11a265,}

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

Vassiliev invariants

V2 and V3: (1, 2)
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 16 8 \frac{62}{3} \frac{34}{3} 64 \frac{256}{3} -\frac{128}{3} 48 \frac{32}{3} 128 \frac{248}{3} \frac{136}{3} \frac{11311}{30} \frac{338}{15} \frac{7262}{45} \frac{689}{18} \frac{271}{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=0 is the signature of K11a56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          2 2
7         51 -4
5        72  5
3       85   -3
1      107    3
-1     89     1
-3    79      -2
-5   48       4
-7  27        -5
-9 14         3
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

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