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

Visit K11n129 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X5,19,6,18 X7,14,8,15 X9,16,10,17 X2,11,3,12 X13,20,14,21 X15,8,16,9 X17,1,18,22 X19,12,20,13 X21,7,22,6
Gauss code 1, -6, 2, -1, -3, 11, -4, 8, -5, -2, 6, 10, -7, 4, -8, 5, -9, 3, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 10 -18 -14 -16 2 -20 -8 -22 -12 -6
A Braid Representative
A Morse Link Presentation K11n129 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n129/ThurstonBennequinNumber
Hyperbolic Volume 11.7204
A-Polynomial See Data:K11n129/A-polynomial

[edit Notes for K11n129'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 K11n129's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3}
Conway polynomial z^6+2 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 43, -2 }
Jones polynomial q-3+5 q^{-1} -6 q^{-2} +8 q^{-3} -7 q^{-4} +6 q^{-5} -4 q^{-6} +2 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -z^4 a^6-3 z^2 a^6-3 a^6+z^6 a^4+5 z^4 a^4+10 z^2 a^4+6 a^4-2 z^4 a^2-5 z^2 a^2-3 a^2+z^2+1
Kauffman polynomial (db, data sources) z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-5 z^4 a^8+z^2 a^8+3 z^7 a^7-9 z^5 a^7+8 z^3 a^7-3 z a^7+3 z^8 a^6-11 z^6 a^6+17 z^4 a^6-11 z^2 a^6+3 a^6+z^9 a^5-7 z^5 a^5+15 z^3 a^5-6 z a^5+4 z^8 a^4-16 z^6 a^4+30 z^4 a^4-21 z^2 a^4+6 a^4+z^9 a^3-3 z^7 a^3+6 z^5 a^3-z^3 a^3-2 z a^3+z^8 a^2-3 z^6 a^2+9 z^4 a^2-11 z^2 a^2+3 a^2+3 z^5 a-5 z^3 a+z^4-2 z^2+1
The A2 invariant -q^{24}-2 q^{20}-q^{18}+2 q^{16}+3 q^{12}+q^{10}+q^8+q^6-2 q^4+q^2-1+ q^{-4}
The G2 invariant Data:K11n129/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_151, K11n54,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_21,}

Vassiliev invariants

V2 and V3: (3, -6)
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
12 -48 72 190 18 -576 -1024 -96 -208 288 1152 2280 216 \frac{52271}{10} -\frac{3946}{15} \frac{33742}{15} \frac{913}{6} \frac{2671}{10}

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 K11n129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
3         11
1        2 -2
-1       31 2
-3      43  -1
-5     42   2
-7    34    1
-9   34     -1
-11  13      2
-13 13       -2
-15 1        1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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