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

Visit K11a301 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X18,5,19,6 X14,7,15,8 X16,10,17,9 X4,11,5,12 X22,13,1,14 X20,16,21,15 X12,18,13,17 X2,19,3,20 X8,21,9,22
Gauss code 1, -10, 2, -6, 3, -1, 4, -11, 5, -2, 6, -9, 7, -4, 8, -5, 9, -3, 10, -8, 11, -7
Dowker-Thistlethwaite code 6 10 18 14 16 4 22 20 12 2 8
A Braid Representative
A Morse Link Presentation K11a301 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a301/ThurstonBennequinNumber
Hyperbolic Volume 19.8329
A-Polynomial See Data:K11a301/A-polynomial

[edit Notes for K11a301's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 2

[edit Notes for K11a301's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+7 t^3-22 t^2+43 t-53+43 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4}
Conway polynomial -z^8-z^6+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 199, -2 }
Jones polynomial q^3-5 q^2+12 q-20+28 q^{-1} -32 q^{-2} +33 q^{-3} -28 q^{-4} +21 q^{-5} -13 q^{-6} +5 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+2 a^4 z^6-4 a^2 z^6+z^6-a^6 z^4+5 a^4 z^4-6 a^2 z^4+2 z^4-a^6 z^2+4 a^4 z^2-2 a^2 z^2+z^2-a^6+a^4+a^2
Kauffman polynomial (db, data sources) 5 a^4 z^{10}+5 a^2 z^{10}+15 a^5 z^9+27 a^3 z^9+12 a z^9+19 a^6 z^8+26 a^4 z^8+18 a^2 z^8+11 z^8+13 a^7 z^7-11 a^5 z^7-47 a^3 z^7-18 a z^7+5 z^7 a^{-1} +5 a^8 z^6-29 a^6 z^6-70 a^4 z^6-59 a^2 z^6+z^6 a^{-2} -22 z^6+a^9 z^5-15 a^7 z^5-13 a^5 z^5+13 a^3 z^5+2 a z^5-8 z^5 a^{-1} -2 a^8 z^4+16 a^6 z^4+46 a^4 z^4+43 a^2 z^4-z^4 a^{-2} +14 z^4+6 a^7 z^3+10 a^5 z^3+5 a^3 z^3+4 a z^3+3 z^3 a^{-1} -5 a^6 z^2-10 a^4 z^2-8 a^2 z^2-3 z^2-2 a^7 z-2 a^5 z+a^6+a^4-a^2
The A2 invariant -q^{24}+2 q^{22}-q^{20}-4 q^{18}+5 q^{16}-5 q^{14}+3 q^{12}+2 q^{10}-3 q^8+7 q^6-6 q^4+6 q^2-1-3 q^{-2} +4 q^{-4} -3 q^{-6} + q^{-8}
The G2 invariant Data:K11a301/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: (2, -3)
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
8 -24 32 \frac{268}{3} \frac{44}{3} -192 -368 -32 -88 \frac{256}{3} 288 \frac{2144}{3} \frac{352}{3} \frac{22951}{15} -\frac{508}{5} \frac{32644}{45} \frac{425}{9} \frac{1351}{15}

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 K11a301. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          4 -4
3         81 7
1        124  -8
-1       168   8
-3      1713    -4
-5     1615     1
-7    1217      5
-9   916       -7
-11  412        8
-13 19         -8
-15 4          4
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=-1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{17} {\mathbb Z}^{17}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\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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