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

Visit K11n10 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X18,11,19,12 X13,6,14,7 X20,15,21,16 X22,17,1,18 X12,19,13,20 X16,21,17,22
Gauss code 1, -5, 2, -1, 3, 7, -4, -2, 5, -3, 6, -10, -7, 4, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 10 -14 2 18 -6 20 22 12 16
A Braid Representative
A Morse Link Presentation K11n10 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n10/ThurstonBennequinNumber
Hyperbolic Volume 13.5825
A-Polynomial See Data:K11n10/A-polynomial

[edit Notes for K11n10'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 K11n10's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-15 t+19-15 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 65, -4 }
Jones polynomial 2 q^{-2} -4 q^{-3} +8 q^{-4} -10 q^{-5} +11 q^{-6} -11 q^{-7} +9 q^{-8} -6 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-a^{10}+2 z^4 a^8+4 z^2 a^8+2 a^8-z^6 a^6-3 z^4 a^6-4 z^2 a^6-3 a^6+2 z^4 a^4+5 z^2 a^4+3 a^4
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-6 z^4 a^{12}+3 z^2 a^{12}+4 z^7 a^{11}-6 z^5 a^{11}+z^3 a^{11}+3 z^8 a^{10}-z^6 a^{10}-4 z^4 a^{10}+a^{10}+z^9 a^9+5 z^7 a^9-9 z^5 a^9+3 z^3 a^9+5 z^8 a^8-7 z^6 a^8+7 z^4 a^8-6 z^2 a^8+2 a^8+z^9 a^7+2 z^7 a^7-z^5 a^7-z^3 a^7+2 z^8 a^6-3 z^6 a^6+8 z^4 a^6-9 z^2 a^6+3 a^6+z^7 a^5+z^5 a^5-z^3 a^5-z a^5+3 z^4 a^4-6 z^2 a^4+3 a^4
The A2 invariant -q^{34}+q^{30}-2 q^{28}+2 q^{26}-q^{22}+q^{20}-3 q^{18}+2 q^{16}-q^{14}+q^{12}+3 q^{10}-q^8+2 q^6
The G2 invariant Data:K11n10/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n103, K11n144,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n144,}

Vassiliev invariants

V2 and V3: (4, -9)
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
16 -72 128 \frac{1160}{3} \frac{160}{3} -1152 -2256 -384 -264 \frac{2048}{3} 2592 \frac{18560}{3} \frac{2560}{3} \frac{203222}{15} \frac{9872}{15} \frac{212528}{45} \frac{778}{9} \frac{8582}{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=-4 is the signature of K11n10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-3         22
-5        31-2
-7       51 4
-9      53  -2
-11     65   1
-13    55    0
-15   46     -2
-17  25      3
-19 14       -3
-21 2        2
-231         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}

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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