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

Visit K11n137 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X5,16,6,17 X7,14,8,15 X18,9,19,10 X20,11,21,12 X2,13,3,14 X15,6,16,7 X22,18,1,17 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -1, -3, 8, -4, -11, 5, -10, 6, -2, 7, 4, -8, 3, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 12 -16 -14 18 20 2 -6 22 10 8
A Braid Representative
A Morse Link Presentation K11n137 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n137/ThurstonBennequinNumber
Hyperbolic Volume 12.3641
A-Polynomial See Data:K11n137/A-polynomial

[edit Notes for K11n137's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n137's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-13 t+15-13 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 57, -4 }
Jones polynomial 1-3 q^{-1} +5 q^{-2} -7 q^{-3} +10 q^{-4} -9 q^{-5} +9 q^{-6} -7 q^{-7} +4 q^{-8} -2 q^{-9}
HOMFLY-PT polynomial (db, data sources) -2 z^2 a^8-3 a^8+3 z^4 a^6+8 z^2 a^6+4 a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4+z^4 a^2+2 z^2 a^2
Kauffman polynomial (db, data sources) 3 z^3 a^{11}-3 z a^{11}+z^6 a^{10}+2 z^4 a^{10}-z^2 a^{10}+2 z^7 a^9+z a^9+2 z^8 a^8-3 z^4 a^8+6 z^2 a^8-3 a^8+z^9 a^7+2 z^7 a^7-2 z^5 a^7-6 z^3 a^7+5 z a^7+5 z^8 a^6-10 z^6 a^6+3 z^4 a^6+3 z^2 a^6-4 a^6+z^9 a^5+3 z^7 a^5-12 z^5 a^5+5 z^3 a^5+z a^5+3 z^8 a^4-8 z^6 a^4+5 z^4 a^4-2 z^2 a^4+3 z^7 a^3-10 z^5 a^3+8 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2
The A2 invariant Data:K11n137/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n137/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n109,}

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

Vassiliev invariants

V2 and V3: (6, -14)
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
24 -112 288 716 132 -2688 -\frac{14752}{3} -\frac{2560}{3} -816 2304 6272 17184 3168 \frac{170711}{5} -\frac{10052}{15} \frac{235724}{15} \frac{841}{3} \frac{11351}{5}

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=-4 is the signature of K11n137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1         11
-1        2 -2
-3       31 2
-5      53  -2
-7     52   3
-9    45    1
-11   55     0
-13  24      2
-15 25       -3
-17 2        2
-192         -2
Integral Khovanov Homology

(db, data source)

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

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See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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