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

Visit K11a288 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,6,17,5 X18,7,19,8 X20,10,21,9 X4,11,5,12 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21
Gauss code 1, -8, 2, -6, 3, -1, 4, -7, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 10 16 18 20 4 8 2 22 14 12
A Braid Representative
A Morse Link Presentation K11a288 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a288/ThurstonBennequinNumber
Hyperbolic Volume 20.2067
A-Polynomial See Data:K11a288/A-polynomial

[edit Notes for K11a288'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 K11a288's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+23 t^2-44 t+55-44 t^{-1} +23 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 { 205, 0 }
Jones polynomial q^6-5 q^5+12 q^4-21 q^3+29 q^2-33 q+34-29 q^{-1} +22 q^{-2} -13 q^{-3} +5 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +4 z^6-2 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +7 z^4-2 a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +6 z^2-a^2- a^{-2} +3
Kauffman polynomial (db, data sources) 5 z^{10} a^{-2} +5 z^{10}+16 a z^9+28 z^9 a^{-1} +12 z^9 a^{-3} +20 a^2 z^8+21 z^8 a^{-2} +11 z^8 a^{-4} +30 z^8+13 a^3 z^7-12 a z^7-45 z^7 a^{-1} -15 z^7 a^{-3} +5 z^7 a^{-5} +5 a^4 z^6-31 a^2 z^6-65 z^6 a^{-2} -21 z^6 a^{-4} +z^6 a^{-6} -79 z^6+a^5 z^5-14 a^3 z^5-13 a z^5+6 z^5 a^{-1} -4 z^5 a^{-3} -8 z^5 a^{-5} -2 a^4 z^4+19 a^2 z^4+47 z^4 a^{-2} +14 z^4 a^{-4} -z^4 a^{-6} +53 z^4+6 a^3 z^3+10 a z^3+8 z^3 a^{-1} +8 z^3 a^{-3} +4 z^3 a^{-5} -6 a^2 z^2-12 z^2 a^{-2} -3 z^2 a^{-4} -15 z^2-a^3 z-2 a z-2 z a^{-1} -z a^{-3} +a^2+ a^{-2} +3
The A2 invariant -q^{14}+3 q^{12}-5 q^{10}+3 q^8+q^6-5 q^4+8 q^2-5+6 q^{-2} - q^{-4} -2 q^{-6} +5 q^{-8} -6 q^{-10} +3 q^{-12} -2 q^{-16} + q^{-18}
The G2 invariant Data:K11a288/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, 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 0 32 -32 \frac{32}{3} 0 \frac{56}{3} -\frac{56}{3} \frac{511}{30} \frac{898}{15} -\frac{1858}{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=0 is the signature of K11a288. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13           11
11          4 -4
9         81 7
7        134  -9
5       168   8
3      1713    -4
1     1716     1
-1    1318      5
-3   916       -7
-5  413        9
-7 19         -8
-9 4          4
-111           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{18}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{17}
r=1 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{17} {\mathbb Z}^{17}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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