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

Visit K11n53 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 X17,20,18,21 X19,11,20,10 X21,12,22,13
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 11, -7, 3, -8, 5, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code 4 8 -14 2 -16 -18 -6 -22 -20 -10 -12
A Braid Representative
A Morse Link Presentation K11n53 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:K11n53/ThurstonBennequinNumber
Hyperbolic Volume 10.3948
A-Polynomial See Data:K11n53/A-polynomial

[edit Notes for K11n53'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 K11n53's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+4 t^2-8 t+11-8 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 { 37, 0 }
Jones polynomial -q^3+3 q^2-4 q+6-6 q^{-1} +6 q^{-2} -5 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6+a^4 z^4-5 a^2 z^4+2 z^4+3 a^4 z^2-9 a^2 z^2-z^2 a^{-2} +6 z^2+2 a^4-5 a^2- a^{-2} +5
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+2 a^5 z^7-a^3 z^7-2 a z^7+z^7 a^{-1} +a^6 z^6-7 a^4 z^6-17 a^2 z^6-9 z^6-8 a^5 z^5-7 a^3 z^5-a z^5-2 z^5 a^{-1} -4 a^6 z^4+6 a^4 z^4+27 a^2 z^4+3 z^4 a^{-2} +20 z^4+8 a^5 z^3+12 a^3 z^3+8 a z^3+5 z^3 a^{-1} +z^3 a^{-3} +3 a^6 z^2-4 a^4 z^2-19 a^2 z^2-4 z^2 a^{-2} -16 z^2-3 a^5 z-6 a^3 z-5 a z-3 z a^{-1} -z a^{-3} +2 a^4+5 a^2+ a^{-2} +5
The A2 invariant Data:K11n53/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n53/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_17,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_14,}

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{34}{3} \frac{62}{3} -64 -\frac{512}{3} -\frac{128}{3} -80 -\frac{32}{3} 128 -\frac{136}{3} -\frac{248}{3} \frac{10289}{30} \frac{154}{5} \frac{2338}{45} \frac{1711}{18} -\frac{751}{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 K11n53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7         1-1
5        2 2
3       21 -1
1      42  2
-1     33   0
-3    33    0
-5   23     1
-7  13      -2
-9 12       1
-11 1        -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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