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

Visit K11a218 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X16,8,17,7 X20,9,21,10 X14,12,15,11 X2,13,3,14 X8,16,9,15 X22,17,1,18 X10,19,11,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -8, 5, -10, 6, -2, 7, -6, 8, -4, 9, -3, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 12 18 16 20 14 2 8 22 10 6
A Braid Representative
A Morse Link Presentation K11a218 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:K11a218/ThurstonBennequinNumber
Hyperbolic Volume 16.744
A-Polynomial See Data:K11a218/A-polynomial

[edit Notes for K11a218'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 K11a218's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-10 t^2+32 t-45+32 t^{-1} -10 t^{-2} + t^{-3}
Conway polynomial z^6-4 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 131, -2 }
Jones polynomial q^3-4 q^2+9 q-14+19 q^{-1} -21 q^{-2} +21 q^{-3} -18 q^{-4} +13 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^8+3 z^2 a^6+2 a^6-3 z^4 a^4-3 z^2 a^4-a^4+z^6 a^2+z^4 a^2+z^2 a^2-2 z^4-z^2+1+z^2 a^{-2}
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+11 a^3 z^9+6 a z^9+6 a^6 z^8+9 a^4 z^8+10 a^2 z^8+7 z^8+5 a^7 z^7-a^5 z^7-18 a^3 z^7-8 a z^7+4 z^7 a^{-1} +3 a^8 z^6-5 a^6 z^6-20 a^4 z^6-29 a^2 z^6+z^6 a^{-2} -16 z^6+a^9 z^5-6 a^7 z^5-6 a^5 z^5+8 a^3 z^5-2 a z^5-9 z^5 a^{-1} -5 a^8 z^4-2 a^6 z^4+11 a^4 z^4+20 a^2 z^4-2 z^4 a^{-2} +10 z^4-2 a^9 z^3+2 a^7 z^3+6 a^5 z^3-3 a^3 z^3+5 z^3 a^{-1} +3 a^8 z^2+6 a^6 z^2+a^4 z^2-6 a^2 z^2+z^2 a^{-2} -3 z^2+a^9 z-a^5 z+a^3 z+a z-a^8-2 a^6-a^4+1
The A2 invariant Data:K11a218/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a218/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a131,}

Vassiliev invariants

V2 and V3: (1, -3)
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
4 -24 8 \frac{350}{3} \frac{106}{3} -96 -464 -32 -184 \frac{32}{3} 288 \frac{1400}{3} \frac{424}{3} \frac{55471}{30} -\frac{9622}{15} \frac{68822}{45} \frac{2129}{18} \frac{7951}{30}

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 K11a218. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          3 -3
3         61 5
1        83  -5
-1       116   5
-3      119    -2
-5     1010     0
-7    811      3
-9   510       -5
-11  28        6
-13 15         -4
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

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