From Knot Atlas
Jump to: navigation, search






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

Visit K11a108 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 X18,15,19,16 X6,18,7,17 X8,19,9,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -11, 7, -3, 8, -4, 9, -8, 10, -6, 11, -7
Dowker-Thistlethwaite code 4 10 14 16 2 20 22 18 6 8 12
A Braid Representative
A Morse Link Presentation K11a108 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:K11a108/ThurstonBennequinNumber
Hyperbolic Volume 14.2368
A-Polynomial See Data:K11a108/A-polynomial

[edit Notes for K11a108's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a108's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-12 t^2+20 t-23+20 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 99, -2 }
Jones polynomial -q^4+3 q^3-6 q^2+10 q-13+16 q^{-1} -15 q^{-2} +14 q^{-3} -11 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-14 a^2 z^4-z^4 a^{-2} +9 z^4+5 a^4 z^2-14 a^2 z^2-3 z^2 a^{-2} +13 z^2+a^4-4 a^2-2 a^{-2} +6
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +7 a^4 z^8+11 a^2 z^8+3 z^8 a^{-2} +7 z^8+7 a^5 z^7+a^3 z^7-14 a z^7-7 z^7 a^{-1} +z^7 a^{-3} +5 a^6 z^6-11 a^4 z^6-38 a^2 z^6-12 z^6 a^{-2} -34 z^6+3 a^7 z^5-9 a^5 z^5-17 a^3 z^5-4 a z^5-3 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-4 a^6 z^4+9 a^4 z^4+41 a^2 z^4+15 z^4 a^{-2} +42 z^4-3 a^7 z^3+5 a^5 z^3+15 a^3 z^3+14 a z^3+12 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+a^6 z^2-5 a^4 z^2-23 a^2 z^2-7 z^2 a^{-2} -23 z^2+a^7 z+a^5 z-3 a^3 z-6 a z-5 z a^{-1} -2 z a^{-3} +a^4+4 a^2+2 a^{-2} +6
The A2 invariant q^{20}-q^{18}+2 q^{16}-2 q^{14}-2 q^{12}+q^{10}-3 q^8+4 q^6-q^4+2 q^2+2- q^{-2} +3 q^{-4} - q^{-6} - q^{-12}
The G2 invariant q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+5 q^{106}-4 q^{104}-2 q^{102}+10 q^{100}-18 q^{98}+26 q^{96}-30 q^{94}+26 q^{92}-11 q^{90}-11 q^{88}+42 q^{86}-67 q^{84}+82 q^{82}-85 q^{80}+58 q^{78}-10 q^{76}-54 q^{74}+123 q^{72}-159 q^{70}+166 q^{68}-121 q^{66}+33 q^{64}+71 q^{62}-169 q^{60}+217 q^{58}-192 q^{56}+95 q^{54}+28 q^{52}-133 q^{50}+183 q^{48}-144 q^{46}+33 q^{44}+91 q^{42}-185 q^{40}+182 q^{38}-89 q^{36}-77 q^{34}+236 q^{32}-310 q^{30}+275 q^{28}-132 q^{26}-74 q^{24}+261 q^{22}-364 q^{20}+344 q^{18}-214 q^{16}+19 q^{14}+174 q^{12}-275 q^{10}+279 q^8-174 q^6+20 q^4+124 q^2-200+173 q^{-2} -59 q^{-4} -86 q^{-6} +209 q^{-8} -233 q^{-10} +159 q^{-12} -12 q^{-14} -145 q^{-16} +257 q^{-18} -275 q^{-20} +196 q^{-22} -61 q^{-24} -85 q^{-26} +184 q^{-28} -207 q^{-30} +165 q^{-32} -82 q^{-34} -3 q^{-36} +61 q^{-38} -88 q^{-40} +76 q^{-42} -48 q^{-44} +18 q^{-46} +4 q^{-48} -15 q^{-50} +14 q^{-52} -11 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a57, K11a139, K11a231,}

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

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{62}{3} \frac{58}{3} 32 -\frac{112}{3} -\frac{160}{3} -24 \frac{32}{3} 32 \frac{248}{3} \frac{232}{3} \frac{8911}{30} \frac{446}{5} \frac{7142}{45} \frac{497}{18} -\frac{209}{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 K11a108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           1-1
7          2 2
5         41 -3
3        62  4
1       74   -3
-1      96    3
-3     78     1
-5    78      -1
-7   47       3
-9  27        -5
-11 14         3
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

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

Back to the top.