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

K11a121 K11a123

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 16 2 20 22 8 12 6

Alexander Polynomial:

2t-3 - 12t-2 + 30t-1 - 39 + 30t - 12t2 + 2t3

Conway Polynomial:

1 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a1, K11a149, ...}

Determinant and Signature:

{127, -2}

Jones Polynomial:

q-9 - 3q-8 + 7q-7 - 13q-6 + 17q-5 - 20q-4 + 21q-3 - 18q-2 + 14q-1 - 8 + 4q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-24 + 3q-22 - 3q-20 - 2q-18 + 2q-16 - 4q-14 + 3q-12 - q-10 + q-8 + 3q-6 - 3q-4 + 5q-2 - 1 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + 2a2 + 3a2z2 + 2a2z4 + a2z6 + a4z4 + a4z6 - 2a6 - 3a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 2z2 - 6z4 + 4z6 + 3az3 - 10az5 + 7az7 - 2a2 + 3a2z2 + a2z4 - 10a2z6 + 8a2z8 + 2a3z + a3z3 - 3a3z5 - 4a3z7 + 6a3z9 - 6a4z2 + 23a4z4 - 27a4z6 + 10a4z8 + 2a4z10 + 2a5z - 9a5z3 + 20a5z5 - 23a5z7 + 11a5z9 + 2a6 - 13a6z2 + 28a6z4 - 26a6z6 + 7a6z8 + 2a6z10 - 2a7z + 4a7z5 - 9a7z7 + 5a7z9 + a8 - 4a8z2 + 9a8z4 - 12a8z6 + 5a8z8 - 2a9z + 6a9z3 - 8a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{0, 2}

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=-2 is the signature of 11122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     3   j = 1                   5 1   j = -1                 9 3     j = -3               10 6       j = -5             11 8         j = -7           9 10           j = -9         8 11             j = -11       5 9               j = -13     2 8                 j = -15   1 5                   j = -17   2                     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, Alternating, 122]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 122]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[18, 7, 19, 8], > X[16, 9, 17, 10], X[2, 11, 3, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[8, 17, 9, 18], X[12, 19, 13, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 122]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -10, 7, -3, 8, -5, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 122]]
Out[5]=	DTCode[4, 10, 14, 18, 16, 2, 20, 22, 8, 12, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 122]][t]
Out[6]=	2 12 30 2 3 -39 + -- - -- + -- + 30 t - 12 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 122]][z]
Out[7]=	6 1 + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 1], Knot[11, Alternating, 122], > Knot[11, Alternating, 149]}
In[9]:=	{KnotDet[Knot[11, Alternating, 122]], KnotSignature[Knot[11, Alternating, 122]]}
Out[9]=	{127, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 122]][q]
Out[10]=	-9 3 7 13 17 20 21 18 14 2 -8 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 q - 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, Alternating, 122]}
In[12]:=	A2Invariant[Knot[11, Alternating, 122]][q]
Out[12]=	-28 -24 3 3 2 2 4 3 -10 -8 3 3 -1 + q - q + --- - --- - --- + --- - --- + --- - q + q + -- - -- + 22 20 18 16 14 12 6 4 q q q q q q q q 5 2 4 6 > -- - q + 2 q - q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 122]][a, z]
Out[13]=	2 6 8 2 2 2 6 2 8 2 4 2 4 4 4 2 a - 2 a + a - z + 3 a z - 3 a z + a z - z + 2 a z + a z - 6 4 2 6 4 6 > 2 a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 122]][a, z]
Out[14]=	2 6 8 3 5 7 9 2 2 2 -2 a + 2 a + a + 2 a z + 2 a z - 2 a z - 2 a z + 2 z + 3 a z - 3 4 2 6 2 8 2 10 2 z 3 3 3 5 3 > 6 a z - 13 a z - 4 a z + 2 a z - -- + 3 a z + a z - 9 a z + a 5 9 3 4 2 4 4 4 6 4 8 4 10 4 z > 6 a z - 6 z + a z + 23 a z + 28 a z + 9 a z - 3 a z + -- - a 5 3 5 5 5 7 5 9 5 6 2 6 > 10 a z - 3 a z + 20 a z + 4 a z - 8 a z + 4 z - 10 a z - 4 6 6 6 8 6 10 6 7 3 7 5 7 > 27 a z - 26 a z - 12 a z + a z + 7 a z - 4 a z - 23 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 9 a z + 3 a z + 8 a z + 10 a z + 7 a z + 5 a z + 6 a z + 5 9 7 9 4 10 6 10 > 11 a z + 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 122]], Vassiliev[3][Knot[11, Alternating, 122]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 122]][q, t]
Out[16]=	6 9 1 2 1 5 2 8 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 9 8 11 9 10 11 8 10 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a122

K11a121 K11a123