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

Visit K11n37 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X2837 X9,19,10,18 X11,6,12,7 X13,20,14,21 X15,22,16,1 X17,11,18,10 X19,16,20,17 X21,14,22,15
Gauss code 1, -4, 2, -1, -3, 6, 4, -2, -5, 9, -6, 3, -7, 11, -8, 10, -9, 5, -10, 7, -11, 8
Dowker-Thistlethwaite code 4 8 -12 2 -18 -6 -20 -22 -10 -16 -14
A Braid Representative
A Morse Link Presentation K11n37 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n37/ThurstonBennequinNumber
Hyperbolic Volume 9.96525
A-Polynomial See Data:K11n37/A-polynomial

[edit Notes for K11n37's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n37's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+3 t^2-5 t+7-5 t^{-1} +3 t^{-2} - t^{-3}
Conway polynomial -z^6-3 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 25, 0 }
Jones polynomial q^2-2 q+4-4 q^{-1} +4 q^{-2} -4 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+z^4+3 a^4 z^2-8 a^2 z^2+3 z^2+2 a^4-4 a^2+3
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+2 a^4 z^8+3 a^2 z^8+z^8+2 a^5 z^7-3 a^3 z^7-5 a z^7+a^6 z^6-8 a^4 z^6-15 a^2 z^6-6 z^6-8 a^5 z^5+8 a z^5-4 a^6 z^4+8 a^4 z^4+25 a^2 z^4+13 z^4+7 a^5 z^3+3 a^3 z^3-3 a z^3+z^3 a^{-1} +3 a^6 z^2-5 a^4 z^2-17 a^2 z^2-9 z^2-2 a^5 z-2 a^3 z+2 a^4+4 a^2+3
The A2 invariant Data:K11n37/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n37/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_9, 10_155,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_137, 10_155,}

Vassiliev invariants

V2 and V3: (-2, 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
-8 16 32 \frac{164}{3} \frac{124}{3} -128 -\frac{800}{3} -\frac{224}{3} -80 -\frac{256}{3} 128 -\frac{1312}{3} -\frac{992}{3} -\frac{31}{15} \frac{3484}{15} -\frac{22684}{45} \frac{1567}{9} -\frac{2431}{15}

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 K11n37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5        11
3       1 -1
1      31 2
-1     22  0
-3    22   0
-5   22    0
-7  12     -1
-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}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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