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

Visit K11a111 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X18,16,19,15 X8,18,9,17 X6,19,7,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 10 14 16 2 22 20 18 8 6 12
A Braid Representative
A Morse Link Presentation K11a111 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a111/ThurstonBennequinNumber
Hyperbolic Volume 13.6745
A-Polynomial See Data:K11a111/A-polynomial

[edit Notes for K11a111's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a111's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-10 t^2+24 t-31+24 t^{-1} -10 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 103, 2 }
Jones polynomial q^7-4 q^6+7 q^5-11 q^4+15 q^3-16 q^2+16 q-13+10 q^{-1} -6 q^{-2} +3 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6-a^2 z^4+2 z^4 a^{-2} -2 z^4 a^{-4} +3 z^4-2 a^2 z^2+2 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} +4 z^2-a^2+ a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+3 a z^9+7 z^9 a^{-1} +4 z^9 a^{-3} +3 a^2 z^8+11 z^8 a^{-2} +7 z^8 a^{-4} +7 z^8+a^3 z^7-7 a z^7-13 z^7 a^{-1} +3 z^7 a^{-3} +8 z^7 a^{-5} -12 a^2 z^6-33 z^6 a^{-2} -5 z^6 a^{-4} +7 z^6 a^{-6} -33 z^6-4 a^3 z^5-3 a z^5-7 z^5 a^{-1} -19 z^5 a^{-3} -7 z^5 a^{-5} +4 z^5 a^{-7} +15 a^2 z^4+24 z^4 a^{-2} -6 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +37 z^4+5 a^3 z^3+12 a z^3+17 z^3 a^{-1} +12 z^3 a^{-3} -z^3 a^{-5} -3 z^3 a^{-7} -7 a^2 z^2-4 z^2 a^{-2} +5 z^2 a^{-4} +2 z^2 a^{-6} -14 z^2-2 a^3 z-5 a z-5 z a^{-1} -z a^{-3} +z a^{-5} +a^2- a^{-2} - a^{-4} +2
The A2 invariant Data:K11a111/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a111/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_117, K11a23,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a68,}

Vassiliev invariants

V2 and V3: (2, 1)
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 8 32 \frac{124}{3} -\frac{4}{3} 64 \frac{368}{3} \frac{128}{3} 8 \frac{256}{3} 32 \frac{992}{3} -\frac{32}{3} \frac{6751}{15} \frac{796}{15} \frac{4804}{45} -\frac{31}{9} \frac{271}{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=2 is the signature of K11a111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
15           11
13          3 -3
11         41 3
9        73  -4
7       84   4
5      87    -1
3     88     0
1    69      3
-1   47       -3
-3  26        4
-5 14         -3
-7 2          2
-91           -1
Integral Khovanov Homology

(db, data source)

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