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

Visit K11n172 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X5,16,6,17 X12,8,13,7 X20,10,21,9 X2,11,3,12 X18,13,19,14 X15,4,16,5 X22,17,1,18 X8,20,9,19 X14,21,15,22
Gauss code 1, -6, 2, 8, -3, -1, 4, -10, 5, -2, 6, -4, 7, -11, -8, 3, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 10 -16 12 20 2 18 -4 22 8 14
A Braid Representative
A Morse Link Presentation K11n172 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:K11n172/ThurstonBennequinNumber
Hyperbolic Volume 13.3915
A-Polynomial See Data:K11n172/A-polynomial

[edit Notes for K11n172's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [0,3]
Rasmussen s-Invariant 0

[edit Notes for K11n172's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+5 t^2-11 t+15-11 t^{-1} +5 t^{-2} - t^{-3}
Conway polynomial -z^6-z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 49, 0 }
Jones polynomial q^2-3 q+6-7 q^{-1} +8 q^{-2} -8 q^{-3} +7 q^{-4} -5 q^{-5} +3 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-a^6+2 z^4 a^4+5 z^2 a^4+3 a^4-z^6 a^2-4 z^4 a^2-6 z^2 a^2-3 a^2+z^4+2 z^2+2
Kauffman polynomial (db, data sources) 2 a^5 z^9+2 a^3 z^9+3 a^6 z^8+8 a^4 z^8+5 a^2 z^8+a^7 z^7-4 a^5 z^7-a^3 z^7+4 a z^7-13 a^6 z^6-32 a^4 z^6-18 a^2 z^6+z^6-4 a^7 z^5-8 a^5 z^5-15 a^3 z^5-11 a z^5+16 a^6 z^4+36 a^4 z^4+22 a^2 z^4+2 z^4+5 a^7 z^3+15 a^5 z^3+17 a^3 z^3+10 a z^3+3 z^3 a^{-1} -6 a^6 z^2-16 a^4 z^2-14 a^2 z^2+z^2 a^{-2} -3 z^2-2 a^7 z-5 a^5 z-5 a^3 z-3 a z-z a^{-1} +a^6+3 a^4+3 a^2+2
The A2 invariant -q^{22}+q^{18}-q^{16}+2 q^{14}+q^8-2 q^6+q^4-q^2+1+2 q^{-2} - q^{-4} + q^{-6}
The G2 invariant Data:K11n172/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_27, K11n4, K11n21,}

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

Vassiliev invariants

V2 and V3: (0, -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
0 -8 0 32 8 0 -\frac{80}{3} -\frac{32}{3} 24 0 32 0 0 32 \frac{760}{3} -\frac{776}{3} -\frac{128}{3} -80

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 K11n172. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5         11
3        2 -2
1       41 3
-1      43  -1
-3     43   1
-5    44    0
-7   34     -1
-9  24      2
-11 13       -2
-13 2        2
-151         -1
Integral Khovanov Homology

(db, data source)

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

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