© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n120 K11n122

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,16,6,17 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,20,14,21 X15,6,16,7 X22,18,1,17 X8,19,9,20 X21,14,22,15

Gauss Code:

{1, -6, 2, -1, -3, 8, -4, -10, 5, -2, 6, 4, -7, 11, -8, 3, 9, -5, 10, 7, -11, -9}

DT (Dowker-Thistlethwaite) Code:

4 10 -16 -12 18 2 -20 -6 22 8 -14

Alexander Polynomial:

- t-3 + 6t-2 - 10t-1 + 11 - 10t + 6t2 - t3

Conway Polynomial:

1 + 5z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n14, ...}

Determinant and Signature:

{45, -4}

Jones Polynomial:

- q-9 + 2q-8 - 5q-7 + 7q-6 - 7q-5 + 8q-4 - 6q-3 + 5q-2 - 3q-1 + 1

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-28 - q-26 - 2q-22 + q-20 + q-16 + 3q-14 + 3q-10 - q-8 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 + a4 - a4z2 - 3a4z4 - a4z6 + 2a6 + 5a6z2 + 2a6z4 - 2a8 - a8z2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 7a3z3 - 10a3z5 + 3a3z7 + a4 - 3a4z2 + 6a4z4 - 9a4z6 + 3a4z8 + a5z + 3a5z3 - 8a5z5 + a5z7 + a5z9 - 2a6 - 2a6z2 + 9a6z4 - 11a6z6 + 4a6z8 + 6a7z - 10a7z3 + 6a7z5 - 2a7z7 + a7z9 - 2a8 + 2a8z2 + 2a8z4 - a8z6 + a8z8 + 4a9z - 5a9z3 + 4a9z5 - a10z2 + 2a10z4 - a11z + a11z3

V2 and V3, the type 2 and 3 Vassiliev invariants:

{5, -11}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-4 is the signature of 11121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                   1 j = -1                 2   j = -3               3 1   j = -5             4 3     j = -7           4 2       j = -9         3 4         j = -11       4 4           j = -13     1 3             j = -15   1 4               j = -17   1                 j = -19 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 121]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 121]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 16, 6, 17], X[7, 12, 8, 13], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[13, 20, 14, 21], X[15, 6, 16, 7], > X[22, 18, 1, 17], X[8, 19, 9, 20], X[21, 14, 22, 15]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 121]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 8, -4, -10, 5, -2, 6, 4, -7, 11, -8, 3, 9, -5, 10, > 7, -11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 121]]
Out[5]=	DTCode[4, 10, -16, -12, 18, 2, -20, -6, 22, 8, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 121]][t]
Out[6]=	-3 6 10 2 3 11 - t + -- - -- - 10 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 121]][z]
Out[7]=	2 6 1 + 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 14], Knot[11, NonAlternating, 121]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 121]], KnotSignature[Knot[11, NonAlternating, 121]]}
Out[9]=	{45, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 121]][q]
Out[10]=	-9 2 5 7 7 8 6 5 3 1 - q + -- - -- + -- - -- + -- - -- + -- - - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 121]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 121]][q]
Out[12]=	-28 -26 2 -20 -16 3 3 -8 -2 1 - q - q - --- + q + q + --- + --- - q - q 22 14 10 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 121]][a, z]
Out[13]=	4 6 8 2 2 4 2 6 2 8 2 2 4 4 4 a + 2 a - 2 a + 2 a z - a z + 5 a z - a z + a z - 3 a z + 6 4 4 6 > 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 121]][a, z]
Out[14]=	4 6 8 5 7 9 11 2 2 4 2 a - 2 a - 2 a + a z + 6 a z + 4 a z - a z + 2 a z - 3 a z - 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 2 a z + 2 a z - a z + 7 a z + 3 a z - 10 a z - 5 a z + 11 3 2 4 4 4 6 4 8 4 10 4 3 5 > a z - 3 a z + 6 a z + 9 a z + 2 a z + 2 a z - 10 a z - 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 8 a z + 6 a z + 4 a z + a z - 9 a z - 11 a z - a z + 3 7 5 7 7 7 4 8 6 8 8 8 5 9 7 9 > 3 a z + a z - 2 a z + 3 a z + 4 a z + a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 121]], Vassiliev[3][Knot[11, NonAlternating, 121]]}
Out[15]=	{5, -11}
In[16]:=	Kh[Knot[11, NonAlternating, 121]][q, t]
Out[16]=	3 3 1 1 1 4 1 3 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 4 3 4 4 2 4 t 2 t 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + q t 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n121

K11n120 K11n122