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

K11a119 K11a121

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,5,15,6 X18,7,19,8 X12,10,13,9 X2,11,3,12 X22,13,1,14 X6,15,7,16 X20,17,21,18 X8,19,9,20 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 12 2 22 6 20 8 16

Alexander Polynomial:

- 2t-3 + 12t-2 - 25t-1 + 31 - 25t + 12t2 - 2t3

Conway Polynomial:

1 + 5z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{109, -4}

Jones Polynomial:

- q-11 + 3q-10 - 7q-9 + 11q-8 - 15q-7 + 18q-6 - 17q-5 + 15q-4 - 11q-3 + 7q-2 - 3q-1 + 1

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

{K11a295, ...}

A2 (sl(3)) Invariant:

- q-34 + q-30 - 3q-28 + q-26 - 2q-22 + 4q-20 - q-18 + 3q-16 - 2q-12 + 3q-10 - 3q-8 + 2q-6 + q-4 - q-2 + 1

HOMFLY-PT Polynomial:

a2 + 2a2z2 + a2z4 - a4 - a4z2 - 2a4z4 - a4z6 + 2a6 + 2a6z2 - a6z4 - a6z6 + 3a8z2 + 2a8z4 - a10 - a10z2

Kauffman Polynomial:

- a2 + 3a2z2 - 3a2z4 + a2z6 - a3z + 6a3z3 - 8a3z5 + 3a3z7 - a4 + 4a4z2 - a4z4 - 7a4z6 + 4a4z8 - 2a5z + 8a5z3 - 11a5z5 + 3a5z9 - 2a6 + 6a6z2 + a6z4 - 14a6z6 + 7a6z8 + a6z10 + 5a7z3 - 7a7z5 - 4a7z7 + 6a7z9 + a8z2 + 6a8z4 - 14a8z6 + 8a8z8 + a8z10 - 2a9z + 10a9z3 - 13a9z5 + 4a9z7 + 3a9z9 + a10 - 2a10z2 + 2a10z4 - 5a10z6 + 5a10z8 - 2a11z + 5a11z3 - 8a11z5 + 5a11z7 + 2a12z2 - 5a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

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

{5, -13}

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 11120. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 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                   5 1   j = -5                 7 3     j = -7               8 4       j = -9             9 7         j = -11           9 8           j = -13         6 9             j = -15       5 9               j = -17     2 6                 j = -19   1 5                   j = -21   2                     j = -23 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, Alternating, 120]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 120]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 5, 15, 6], X[18, 7, 19, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[22, 13, 1, 14], X[6, 15, 7, 16], > X[20, 17, 21, 18], X[8, 19, 9, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 120]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, > -9, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 120]]
Out[5]=	DTCode[4, 10, 14, 18, 12, 2, 22, 6, 20, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 120]][t]
Out[6]=	2 12 25 2 3 31 - -- + -- - -- - 25 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 120]][z]
Out[7]=	2 6 1 + 5 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 120]}
In[9]:=	{KnotDet[Knot[11, Alternating, 120]], KnotSignature[Knot[11, Alternating, 120]]}
Out[9]=	{109, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 120]][q]
Out[10]=	-11 3 7 11 15 18 17 15 11 7 3 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q 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, Alternating, 120], Knot[11, Alternating, 295]}
In[12]:=	A2Invariant[Knot[11, Alternating, 120]][q]
Out[12]=	-34 -30 3 -26 2 4 -18 3 2 3 3 2 1 - q + q - --- + q - --- + --- - q + --- - --- + --- - -- + -- + 28 22 20 16 12 10 8 6 q q q q q q q q -4 -2 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 120]][a, z]
Out[13]=	2 4 6 10 2 2 4 2 6 2 8 2 10 2 2 4 a - a + 2 a - a + 2 a z - a z + 2 a z + 3 a z - a z + a z - 4 4 6 4 8 4 4 6 6 6 > 2 a z - a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 120]][a, z]
Out[14]=	2 4 6 10 3 5 9 11 13 2 2 -a - a - 2 a + a - a z - 2 a z - 2 a z - 2 a z + a z + 3 a z + 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 4 a z + 6 a z + a z - 2 a z + 2 a z + 6 a z + 8 a z + 7 3 9 3 11 3 13 3 2 4 4 4 6 4 > 5 a z + 10 a z + 5 a z - 2 a z - 3 a z - a z + a z + 8 4 10 4 12 4 3 5 5 5 7 5 9 5 > 6 a z + 2 a z - 5 a z - 8 a z - 11 a z - 7 a z - 13 a z - 11 5 13 5 2 6 4 6 6 6 8 6 10 6 > 8 a z + a z + a z - 7 a z - 14 a z - 14 a z - 5 a z + 12 6 3 7 7 7 9 7 11 7 4 8 6 8 > 3 a z + 3 a z - 4 a z + 4 a z + 5 a z + 4 a z + 7 a z + 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 8 a z + 5 a z + 3 a z + 6 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 120]], Vassiliev[3][Knot[11, Alternating, 120]]}
Out[15]=	{5, -13}
In[16]:=	Kh[Knot[11, Alternating, 120]][q, t]
Out[16]=	3 5 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 9 6 9 9 8 9 7 8 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 7 t 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a120

K11a119 K11a121