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

Visit K11a183 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,9,19,10 X20,11,21,12 X2,13,3,14 X22,16,1,15 X10,17,11,18 X8,19,9,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -4, 9, -5, 10, -6, 11, -8
Dowker-Thistlethwaite code 4 12 14 16 18 20 2 22 10 8 6
A Braid Representative
A Morse Link Presentation K11a183 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:K11a183/ThurstonBennequinNumber
Hyperbolic Volume 14.4304
A-Polynomial See Data:K11a183/A-polynomial

[edit Notes for K11a183'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 K11a183'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^2+4 q-8+13 q^{-1} -16 q^{-2} +19 q^{-3} -18 q^{-4} +15 q^{-5} -11 q^{-6} +6 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-2 z^4 a^6-4 z^2 a^6-3 a^6+z^6 a^4+2 z^4 a^4+3 z^2 a^4+2 a^4+z^6 a^2+2 z^4 a^2+2 z^2 a^2+a^2-z^4-z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-9 z^5 a^9+8 z^3 a^9-2 z a^9+4 z^8 a^8-9 z^6 a^8+5 z^4 a^8-z^2 a^8+a^8+3 z^9 a^7-z^7 a^7-9 z^5 a^7+10 z^3 a^7-4 z a^7+z^{10} a^6+8 z^8 a^6-21 z^6 a^6+19 z^4 a^6-10 z^2 a^6+3 a^6+7 z^9 a^5-8 z^7 a^5+2 z^3 a^5-z a^5+z^{10} a^4+11 z^8 a^4-23 z^6 a^4+17 z^4 a^4-7 z^2 a^4+2 a^4+4 z^9 a^3+3 z^7 a^3-12 z^5 a^3+5 z^3 a^3+z a^3+7 z^8 a^2-8 z^6 a^2+2 z^2 a^2-a^2+7 z^7 a-11 z^5 a+4 z^3 a+4 z^6-6 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-1}
The A2 invariant Data:K11a183/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a183/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_121, K11a41, K11a198, K11a331,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a41,}

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{82}{3} -\frac{14}{3} 0 128 -32 64 \frac{32}{3} 0 -\frac{328}{3} -\frac{56}{3} -\frac{12449}{30} \frac{1286}{5} -\frac{13378}{45} -\frac{1663}{18} -\frac{1409}{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 K11a183. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5           1-1
3          3 3
1         51 -4
-1        83  5
-3       96   -3
-5      107    3
-7     89     1
-9    710      -3
-11   48       4
-13  27        -5
-15 14         3
-17 2          -2
-191           1
Integral Khovanov Homology

(db, data source)

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