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

Visit K11n62 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,20,12,21 X13,18,14,19 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 7, -10, 6, -11, 9
Dowker-Thistlethwaite code 4 8 14 2 -16 -20 -18 6 -22 -12 -10
A Braid Representative
A Morse Link Presentation K11n62 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n62/ThurstonBennequinNumber
Hyperbolic Volume 9.61446
A-Polynomial See Data:K11n62/A-polynomial

[edit Notes for K11n62's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11n62's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^2-8 t+13-8 t^{-1} +2 t^{-2}
Conway polynomial 2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 0 }
Jones polynomial -q^5+2 q^4-3 q^3+5 q^2-5 q+6-5 q^{-1} +3 q^{-2} -2 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-2 z^2 a^2-2 a^2+z^4+z^2+1+z^4 a^{-2} +2 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4} - a^{-4}
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +a^2 z^8+2 z^8 a^{-2} +3 z^8-5 a z^7-3 z^7 a^{-1} +2 z^7 a^{-3} -5 a^2 z^6-7 z^6 a^{-2} +2 z^6 a^{-4} -14 z^6+2 a^3 z^5+11 a z^5+3 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +a^4 z^4+12 a^2 z^4+8 z^4 a^{-2} -6 z^4 a^{-4} +25 z^4-5 a^3 z^3-8 a z^3+z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} -3 a^4 z^2-11 a^2 z^2-z^2 a^{-2} +3 z^2 a^{-4} -12 z^2+2 a^3 z+3 a z+z a^{-1} +z a^{-3} +z a^{-5} +a^4+2 a^2-2 a^{-2} - a^{-4} +1
The A2 invariant Data:K11n62/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n62/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_146, K11n18,}

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

Vassiliev invariants

V2 and V3: (0, 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
0 16 0 -16 -16 0 \frac{64}{3} -\frac{128}{3} 48 0 128 0 0 216 \frac{208}{3} \frac{160}{3} -\frac{104}{3} -24

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 K11n62. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         1-1
9        1 1
7       21 -1
5      31  2
3     22   0
1    43    1
-1   23     1
-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=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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