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

Visit K11n58 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code 4 8 14 2 -16 -18 -20 6 -22 -12 -10
A Braid Representative
A Morse Link Presentation K11n58 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n58/ThurstonBennequinNumber
Hyperbolic Volume 9.97833
A-Polynomial See Data:K11n58/A-polynomial

[edit Notes for K11n58'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 K11n58's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-4 t^2+8 t-9+8 t^{-1} -4 t^{-2} + t^{-3}
Conway polynomial z^6+2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 35, -2 }
Jones polynomial -q^4+2 q^3-3 q^2+5 q-5+6 q^{-1} -5 q^{-2} +4 q^{-3} -3 q^{-4} + q^{-5}
HOMFLY-PT polynomial (db, data sources) z^6-2 a^2 z^4-z^4 a^{-2} +5 z^4+a^4 z^2-6 a^2 z^2-3 z^2 a^{-2} +9 z^2+a^4-4 a^2-2 a^{-2} +6
Kauffman polynomial (db, data sources) a z^9+z^9 a^{-1} +2 a^2 z^8+2 z^8 a^{-2} +4 z^8+2 a^3 z^7-a z^7-2 z^7 a^{-1} +z^7 a^{-3} +a^4 z^6-6 a^2 z^6-10 z^6 a^{-2} -17 z^6-5 a^3 z^5-5 a z^5-5 z^5 a^{-1} -5 z^5 a^{-3} +8 a^2 z^4+15 z^4 a^{-2} +23 z^4+3 a^5 z^3+6 a^3 z^3+6 a z^3+10 z^3 a^{-1} +7 z^3 a^{-3} +a^6 z^2-10 a^2 z^2-8 z^2 a^{-2} -17 z^2-a^5 z-3 a^3 z-4 a z-4 z a^{-1} -2 z a^{-3} +a^4+4 a^2+2 a^{-2} +6
The A2 invariant q^{16}-q^{12}-2 q^8+q^2+3+ q^{-2} +2 q^{-4} - q^{-8} - q^{-12}
The G2 invariant Data:K11n58/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_16, 10_156, K11n15, K11n56,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n56,}

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{176}{3} -\frac{64}{3} 40 \frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} -\frac{209}{30} \frac{1258}{15} -\frac{3898}{45} -\frac{463}{18} -\frac{1169}{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 K11n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9         1-1
7        1 1
5       21 -1
3      31  2
1     22   0
-1    43    1
-3   23     1
-5  23      -1
-7 12       1
-9 2        -2
-111         1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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