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

Visit K11a70 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8394 X18,6,19,5 X12,8,13,7 X2,9,3,10 X16,11,17,12 X20,14,21,13 X6,16,7,15 X10,17,11,18 X22,20,1,19 X14,22,15,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -9, 6, -4, 7, -11, 8, -6, 9, -3, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 8 18 12 2 16 20 6 10 22 14
A Braid Representative
A Morse Link Presentation K11a70 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a70/ThurstonBennequinNumber
Hyperbolic Volume 17.4335
A-Polynomial See Data:K11a70/A-polynomial

[edit Notes for K11a70'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 K11a70's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-14 t^2+36 t-47+36 t^{-1} -14 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 { 151, 2 }
Jones polynomial -q^8+5 q^7-11 q^6+17 q^5-22 q^4+25 q^3-24 q^2+20 q-14+8 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -2 z^2+a^2- a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +5 z^9 a^{-1} +13 z^9 a^{-3} +8 z^9 a^{-5} +12 z^8 a^{-2} +20 z^8 a^{-4} +13 z^8 a^{-6} +5 z^8+3 a z^7-4 z^7 a^{-1} -16 z^7 a^{-3} +2 z^7 a^{-5} +11 z^7 a^{-7} +a^2 z^6-33 z^6 a^{-2} -46 z^6 a^{-4} -18 z^6 a^{-6} +5 z^6 a^{-8} -9 z^6-7 a z^5-5 z^5 a^{-1} -4 z^5 a^{-3} -22 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+28 z^4 a^{-2} +30 z^4 a^{-4} +5 z^4 a^{-6} -4 z^4 a^{-8} +4 z^4+5 a z^3+5 z^3 a^{-1} +8 z^3 a^{-3} +12 z^3 a^{-5} +4 z^3 a^{-7} +3 a^2 z^2-10 z^2 a^{-2} -7 z^2 a^{-4} -a z-z a^{-1} -z a^{-3} -z a^{-5} -a^2+ a^{-2} + a^{-4}
The A2 invariant Data:K11a70/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a70/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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{164}{3} \frac{100}{3} 64 \frac{400}{3} \frac{64}{3} 56 -\frac{256}{3} 32 -\frac{1312}{3} -\frac{800}{3} -\frac{4111}{15} \frac{268}{5} -\frac{15364}{45} \frac{655}{9} -\frac{1471}{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 K11a70. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
17           1-1
15          4 4
13         71 -6
11        104  6
9       127   -5
7      1310    3
5     1112     1
3    913      -4
1   612       6
-1  28        -6
-3 16         5
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\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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