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

Visit K11n87 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X20,14,21,13 X8,15,9,16 X17,22,18,1 X6,19,7,20 X21,16,22,17
Gauss code 1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -4, 8, 11, -9, -5, 10, -7, -11, 9
Dowker-Thistlethwaite code 4 10 12 14 18 2 20 8 -22 6 -16
A Braid Representative
A Morse Link Presentation K11n87 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:K11n87/ThurstonBennequinNumber
Hyperbolic Volume 12.3434
A-Polynomial See Data:K11n87/A-polynomial

[edit Notes for K11n87'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 K11n87's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+12 t-15+12 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 51, -2 }
Jones polynomial -1+4 q^{-1} -5 q^{-2} +8 q^{-3} -9 q^{-4} +8 q^{-5} -7 q^{-6} +5 q^{-7} -3 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-2 z^4 a^6-5 z^2 a^6-3 a^6+z^6 a^4+4 z^4 a^4+6 z^2 a^4+2 a^4-z^4 a^2-z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}+3 z^7 a^9-10 z^5 a^9+8 z^3 a^9-2 z a^9+3 z^8 a^8-8 z^6 a^8+3 z^4 a^8+a^8+z^9 a^7+3 z^7 a^7-16 z^5 a^7+14 z^3 a^7-4 z a^7+5 z^8 a^6-14 z^6 a^6+14 z^4 a^6-9 z^2 a^6+3 a^6+z^9 a^5+z^7 a^5-5 z^5 a^5+5 z^3 a^5-z a^5+2 z^8 a^4-5 z^6 a^4+12 z^4 a^4-10 z^2 a^4+2 a^4+z^7 a^3+z^5 a^3+z a^3+4 z^4 a^2-3 z^2 a^2-a^2+z^3 a
The A2 invariant q^{28}-q^{24}+q^{22}-2 q^{20}-2 q^{14}+q^{12}-q^{10}+3 q^8+2 q^6+2 q^2-1
The G2 invariant Data:K11n87/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_28, 9_29, 10_163,}

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{82}{3} -\frac{14}{3} 0 128 0 32 \frac{32}{3} 0 -\frac{328}{3} -\frac{56}{3} -\frac{12449}{30} \frac{326}{5} -\frac{7618}{45} -\frac{511}{18} -\frac{449}{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 K11n87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
1         1-1
-1        3 3
-3       32 -1
-5      52  3
-7     43   -1
-9    45    -1
-11   34     1
-13  24      -2
-15 13       2
-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}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\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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