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

Visit K11n56 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X5,14,6,15 X2837 X9,16,10,17 X11,19,12,18 X13,6,14,7 X15,22,16,1 X17,21,18,20 X19,13,20,12 X21,10,22,11
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code 4 8 -14 2 -16 -18 -6 -22 -20 -12 -10
A Braid Representative
A Morse Link Presentation K11n56 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:K11n56/ThurstonBennequinNumber
Hyperbolic Volume 10.3372
A-Polynomial See Data:K11n56/A-polynomial

[edit Notes for K11n56'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 K11n56's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3}
Conway polynomial z^6+2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, 2 }
Jones polynomial q^5-3 q^4+4 q^3-5 q^2+6 q-5+5 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6-a^2 z^4-2 z^4 a^{-2} +5 z^4-3 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} +9 z^2-2 a^2-4 a^{-2} + a^{-4} +6
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+a^3 z^7-a z^7+z^7 a^{-1} +3 z^7 a^{-3} -10 a^2 z^6-12 z^6 a^{-2} +z^6 a^{-4} -23 z^6-5 a^3 z^5-11 a z^5-17 z^5 a^{-1} -11 z^5 a^{-3} +15 a^2 z^4+17 z^4 a^{-2} +32 z^4+7 a^3 z^3+19 a z^3+24 z^3 a^{-1} +15 z^3 a^{-3} +3 z^3 a^{-5} -9 a^2 z^2-13 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -20 z^2-3 a^3 z-8 a z-10 z a^{-1} -7 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6
The A2 invariant Data:K11n56/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n56/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_16, 10_156, K11n15, K11n58,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n58,}

Vassiliev invariants

V2 and V3: (1, -1)
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 -8 8 -\frac{34}{3} -\frac{38}{3} -32 -\frac{80}{3} -\frac{32}{3} 24 \frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} \frac{751}{30} \frac{58}{15} \frac{422}{45} -\frac{175}{18} -\frac{689}{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 K11n56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         11
9        2 -2
7       21 1
5      32  -1
3     32   1
1    34    1
-1   22     0
-3  13      2
-5 12       -1
-7 1        1
-91         -1
Integral Khovanov Homology

(db, data source)

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