K11a7

K11a6

K11a8

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a7 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X6,16,7,15 X22,17,1,18 X12,19,13,20 X14,21,15,22
Gauss code	1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -10, 7, -11, 8, -4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code	4 8 10 16 2 18 20 6 22 12 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a7/ThurstonBennequinNumber Hyperbolic Volume 13.5338 A-Polynomial See Data:K11a7/A-polynomial

[edit Notes for K11a7's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 2

[edit Notes for K11a7's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-3 z^6-2 z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 95, -2 }
Jones polynomial	[math]\displaystyle{ q^3-3 q^2+6 q-9+13 q^{-1} -15 q^{-2} +15 q^{-3} -13 q^{-4} +10 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-14 a^2 z^4+4 z^4-3 a^6 z^2+13 a^4 z^2-15 a^2 z^2+5 z^2-2 a^6+6 a^4-6 a^2+3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+7 a^3 z^9+4 a z^9+4 a^6 z^8+7 a^4 z^8+8 a^2 z^8+5 z^8+4 a^7 z^7-2 a^5 z^7-19 a^3 z^7-10 a z^7+3 z^7 a^{-1} +3 a^8 z^6-4 a^6 z^6-26 a^4 z^6-37 a^2 z^6+z^6 a^{-2} -17 z^6+a^9 z^5-5 a^7 z^5+a^5 z^5+21 a^3 z^5+5 a z^5-9 z^5 a^{-1} -6 a^8 z^4+a^6 z^4+41 a^4 z^4+55 a^2 z^4-3 z^4 a^{-2} +18 z^4-2 a^9 z^3-a^7 z^3-2 a^5 z^3-6 a^3 z^3+2 a z^3+5 z^3 a^{-1} +3 a^8 z^2-4 a^6 z^2-27 a^4 z^2-31 a^2 z^2+z^2 a^{-2} -10 z^2+a^9 z+a^7 z-a z-z a^{-1} +2 a^6+6 a^4+6 a^2+3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{24}-q^{18}+3 q^{16}-q^{14}+2 q^{12}+q^{10}-2 q^8+2 q^6-4 q^4+2 q^2+2 q^{-4} - q^{-6} + q^{-8} }[/math]
The G2 invariant	[math]\displaystyle{ q^{128}-2 q^{126}+5 q^{124}-8 q^{122}+8 q^{120}-6 q^{118}-2 q^{116}+16 q^{114}-29 q^{112}+41 q^{110}-42 q^{108}+27 q^{106}-2 q^{104}-37 q^{102}+75 q^{100}-101 q^{98}+101 q^{96}-76 q^{94}+20 q^{92}+47 q^{90}-112 q^{88}+160 q^{86}-165 q^{84}+125 q^{82}-45 q^{80}-57 q^{78}+139 q^{76}-178 q^{74}+161 q^{72}-85 q^{70}-16 q^{68}+105 q^{66}-137 q^{64}+101 q^{62}-3 q^{60}-103 q^{58}+170 q^{56}-158 q^{54}+63 q^{52}+80 q^{50}-214 q^{48}+290 q^{46}-259 q^{44}+139 q^{42}+36 q^{40}-202 q^{38}+298 q^{36}-296 q^{34}+199 q^{32}-49 q^{30}-102 q^{28}+199 q^{26}-211 q^{24}+142 q^{22}-22 q^{20}-97 q^{18}+151 q^{16}-137 q^{14}+42 q^{12}+80 q^{10}-175 q^8+207 q^6-153 q^4+34 q^2+104-206 q^{-2} +234 q^{-4} -184 q^{-6} +82 q^{-8} +34 q^{-10} -121 q^{-12} +160 q^{-14} -140 q^{-16} +92 q^{-18} -25 q^{-20} -26 q^{-22} +52 q^{-24} -58 q^{-26} +45 q^{-28} -26 q^{-30} +10 q^{-32} +4 q^{-34} -9 q^{-36} +8 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46} }[/math]

Further Quantum Invariants

K11a7 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a7"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -t^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-3 z^6-2 z^4+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 95, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^3-3 q^2+6 q-9+13 q^{-1} -15 q^{-2} +15 q^{-3} -13 q^{-4} +10 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-14 a^2 z^4+4 z^4-3 a^6 z^2+13 a^4 z^2-15 a^2 z^2+5 z^2-2 a^6+6 a^4-6 a^2+3 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+7 a^3 z^9+4 a z^9+4 a^6 z^8+7 a^4 z^8+8 a^2 z^8+5 z^8+4 a^7 z^7-2 a^5 z^7-19 a^3 z^7-10 a z^7+3 z^7 a^{-1} +3 a^8 z^6-4 a^6 z^6-26 a^4 z^6-37 a^2 z^6+z^6 a^{-2} -17 z^6+a^9 z^5-5 a^7 z^5+a^5 z^5+21 a^3 z^5+5 a z^5-9 z^5 a^{-1} -6 a^8 z^4+a^6 z^4+41 a^4 z^4+55 a^2 z^4-3 z^4 a^{-2} +18 z^4-2 a^9 z^3-a^7 z^3-2 a^5 z^3-6 a^3 z^3+2 a z^3+5 z^3 a^{-1} +3 a^8 z^2-4 a^6 z^2-27 a^4 z^2-31 a^2 z^2+z^2 a^{-2} -10 z^2+a^9 z+a^7 z-a z-z a^{-1} +2 a^6+6 a^4+6 a^2+3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a33, K11a82,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11a325,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a7"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ -t^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^3-3 q^2+6 q-9+13 q^{-1} -15 q^{-2} +15 q^{-3} -13 q^{-4} +10 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{10_116, K11a33, K11a82,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a325,}

Vassiliev invariants

V2 and V3:

(0, -2)

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 [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{160}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{560}{3} }[/math] [math]\displaystyle{ -\frac{976}{3} }[/math] [math]\displaystyle{ \frac{128}{3} }[/math] [math]\displaystyle{ -64 }[/math]

V_2,1 through V_6,9 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of K11a7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   4 1   3 1                 5 2     -3 -1               8 4       4 -3             8 6         -2 -5           7 7           0 -7         6 8             2 -9       4 7               -3 -11     2 6                 4 -13   1 4                   -3 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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.

K11a6 K11a8