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

Visit K11a287 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,6,17,5 X18,7,19,8 X20,10,21,9 X2,11,3,12 X8,13,9,14 X4,16,5,15 X22,17,1,18 X12,20,13,19 X14,21,15,22
Gauss code 1, -6, 2, -8, 3, -1, 4, -7, 5, -2, 6, -10, 7, -11, 8, -3, 9, -4, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 10 16 18 20 2 8 4 22 12 14
A Braid Representative
A Morse Link Presentation K11a287 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a287/ThurstonBennequinNumber
Hyperbolic Volume 19.3323
A-Polynomial See Data:K11a287/A-polynomial

[edit Notes for K11a287's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a287's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+21 t^2-38 t+47-38 t^{-1} +21 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 181, 0 }
Jones polynomial -q^5+5 q^4-12 q^3+20 q^2-26 q+30-29 q^{-1} +25 q^{-2} -18 q^{-3} +10 q^{-4} -4 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) z^8-2 a^2 z^6-z^6 a^{-2} +4 z^6+a^4 z^4-6 a^2 z^4-2 z^4 a^{-2} +6 z^4+2 a^4 z^2-6 a^2 z^2-z^2 a^{-2} +4 z^2+a^4-2 a^2+2
Kauffman polynomial (db, data sources) 4 a^2 z^{10}+4 z^{10}+9 a^3 z^9+22 a z^9+13 z^9 a^{-1} +8 a^4 z^8+14 a^2 z^8+17 z^8 a^{-2} +23 z^8+4 a^5 z^7-13 a^3 z^7-39 a z^7-10 z^7 a^{-1} +12 z^7 a^{-3} +a^6 z^6-16 a^4 z^6-47 a^2 z^6-26 z^6 a^{-2} +5 z^6 a^{-4} -61 z^6-8 a^5 z^5+2 a^3 z^5+16 a z^5-9 z^5 a^{-1} -14 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+12 a^4 z^4+41 a^2 z^4+13 z^4 a^{-2} -3 z^4 a^{-4} +43 z^4+5 a^5 z^3+3 a^3 z^3-a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +a^6 z^2-5 a^4 z^2-15 a^2 z^2-3 z^2 a^{-2} -12 z^2-a^5 z-a^3 z+a^4+2 a^2+2
The A2 invariant q^{18}-q^{16}+3 q^{12}-5 q^{10}+4 q^8-2 q^6-2 q^4+5 q^2-5+7 q^{-2} -4 q^{-4} + q^{-6} +3 q^{-8} -4 q^{-10} +3 q^{-12} - q^{-14}
The G2 invariant Data:K11a287/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: (-1, 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
-4 8 8 \frac{34}{3} \frac{38}{3} -32 -\frac{208}{3} -\frac{160}{3} 8 -\frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} \frac{2129}{30} \frac{1142}{15} -\frac{1862}{45} -\frac{113}{18} -\frac{751}{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=0 is the signature of K11a287. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          4 4
7         81 -7
5        124  8
3       148   -6
1      1612    4
-1     1415     1
-3    1115      -4
-5   714       7
-7  311        -8
-9 17         6
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\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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