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

Visit K11a198 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,5,17,6 X14,7,15,8 X18,10,19,9 X2,12,3,11 X22,13,1,14 X6,15,7,16 X20,18,21,17 X10,20,11,19 X8,21,9,22
Gauss code 1, -6, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -5, 10, -9, 11, -7
Dowker-Thistlethwaite code 4 12 16 14 18 2 22 6 20 10 8
A Braid Representative
A Morse Link Presentation K11a198 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a198/ThurstonBennequinNumber
Hyperbolic Volume 15.066
A-Polynomial See Data:K11a198/A-polynomial

[edit Notes for K11a198'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 K11a198's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+27 t-35+27 t^{-1} -11 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 115, 2 }
Jones polynomial q^7-4 q^6+8 q^5-13 q^4+17 q^3-18 q^2+18 q-15+11 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+z^4 a^{-2} -2 z^4 a^{-4} +3 z^4-2 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} +5 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+3 a z^9+8 z^9 a^{-1} +5 z^9 a^{-3} +3 a^2 z^8+16 z^8 a^{-2} +10 z^8 a^{-4} +9 z^8+a^3 z^7-6 a z^7-12 z^7 a^{-1} +6 z^7 a^{-3} +11 z^7 a^{-5} -12 a^2 z^6-47 z^6 a^{-2} -12 z^6 a^{-4} +8 z^6 a^{-6} -39 z^6-4 a^3 z^5-5 a z^5-15 z^5 a^{-1} -32 z^5 a^{-3} -14 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+38 z^4 a^{-2} +z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +45 z^4+5 a^3 z^3+14 a z^3+25 z^3 a^{-1} +24 z^3 a^{-3} +6 z^3 a^{-5} -2 z^3 a^{-7} -8 a^2 z^2-12 z^2 a^{-2} +z^2 a^{-4} +2 z^2 a^{-6} -19 z^2-2 a^3 z-6 a z-8 z a^{-1} -5 z a^{-3} -z a^{-5} +a^2+ a^{-2} +3
The A2 invariant Data:K11a198/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a198/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_121, K11a41, K11a183, K11a331,}

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

Vassiliev invariants

V2 and V3: (1, 0)
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 0 8 \frac{14}{3} -\frac{14}{3} 0 32 32 0 \frac{32}{3} 0 \frac{56}{3} -\frac{56}{3} \frac{1471}{30} -\frac{302}{15} \frac{2462}{45} -\frac{223}{18} \frac{991}{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 K11a198. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
15           11
13          3 -3
11         51 4
9        83  -5
7       95   4
5      98    -1
3     99     0
1    710      3
-1   48       -4
-3  27        5
-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}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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