The Non Alternating Knot 10₁₄₅

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_5,12,6,13 X₈₃₉₄ X_2,9,3,10 X_11,16,12,17 X_17,10,18,11 X_7,18,8,19 X_13,20,14,1 X_19,14,20,15 X_15,6,16,7

Gauss Code: {1, -4, 3, -1, -2, 10, -7, -3, 4, 6, -5, 2, -8, 9, -10, 5, -6, 7, -9, 8}

DT (Dowker-Thistlethwaite) Code: 4 8 -12 -18 2 -16 -20 -6 -10 -14

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 2 2 3 / NotAvailable 1

Alexander Polynomial: t^-2 + t^-1 - 3 + t + t²

Conway Polynomial: 1 + 5z² + z⁴

Other knots with the same Alexander/Conway Polynomial: {...}

Determinant and Signature: {3, -2}

Jones Polynomial: - q^-10 + q^-9 - q^-8 + q^-7 + q^-2

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

A2 (sl(3)) Invariant: - q^-32 - q^-30 + q^-24 + q^-14 + q^-10 + q^-8 + q^-6

HOMFLY-PT Polynomial: 2a⁴ + 4a⁴z² + a⁴z⁴ - a⁶ + a⁸ + a⁸z² - a¹⁰

Kauffman Polynomial: 2a⁴ - 4a⁴z² + a⁴z⁴ - a⁵z + a⁶ - 2a⁶z² - 2a⁷z + 8a⁷z³ - 6a⁷z⁵ + a⁷z⁷ + a⁸ - 4a⁸z² + 9a⁸z⁴ - 6a⁸z⁶ + a⁸z⁸ - 6a⁹z + 18a⁹z³ - 12a⁹z⁵ + 2a⁹z⁷ + a¹⁰ - 6a¹⁰z² + 10a¹⁰z⁴ - 6a¹⁰z⁶ + a¹⁰z⁸ - 5a¹¹z + 10a¹¹z³ - 6a¹¹z⁵ + a¹¹z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {5, -12}

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

t^rq^j r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -3 1

j = -5 1

j = -7 1 1

j = -9

j = -11 1 2 1

j = -13 1

j = -15 1 1

j = -17 1 1

j = -19

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-29 - q^-28 - q^-27 + 2q^-26 - q^-25 - q^-24 + 2q^-23 - 2q^-22 + q^-21 + 2q^-20 - 4q^-19 + 2q^-18 + 2q^-17 - 5q^-16 + 2q^-15 + 2q^-14 - 5q^-13 + 4q^-12 + 2q^-11 - 4q^-10 + 2q^-9 + q^-8 - q^-7 + q^-4

3 - q^-57 + q^-56 + q^-55 - 2q^-53 + 2q^-51 - 2q^-49 + 3q^-47 - 2q^-46 - 3q^-45 + 2q^-44 + 5q^-43 - 2q^-42 - 6q^-41 + 2q^-40 + 6q^-39 - q^-38 - 5q^-37 + 4q^-35 - 3q^-33 + q^-31 + q^-30 - q^-29 - q^-28 + q^-26 + q^-25 - 2q^-24 - q^-23 + q^-22 + 2q^-21 - 2q^-20 - 2q^-19 + q^-18 + 3q^-17 - 2q^-15 - q^-14 + 2q^-13 + q^-12 - q^-10 + q^-6

4 q^-94 - q^-93 - q^-92 + 3q^-89 - q^-88 - q^-87 - q^-86 - q^-85 + 4q^-84 - q^-83 - q^-82 - 2q^-81 + q^-80 + 5q^-79 - 2q^-78 - 3q^-77 - 5q^-76 + 3q^-75 + 7q^-74 + q^-73 - 3q^-72 - 8q^-71 + 2q^-70 + 6q^-69 + q^-68 - 2q^-67 - 4q^-66 + 3q^-65 + 4q^-64 - 3q^-63 - 5q^-62 - q^-61 + 6q^-60 + 8q^-59 - 6q^-58 - 10q^-57 - q^-56 + 7q^-55 + 13q^-54 - 6q^-53 - 12q^-52 - 3q^-51 + 5q^-50 + 18q^-49 - 5q^-48 - 14q^-47 - 5q^-46 + 4q^-45 + 21q^-44 - 5q^-43 - 16q^-42 - 4q^-41 + 3q^-40 + 21q^-39 - 6q^-38 - 16q^-37 - 2q^-36 + 3q^-35 + 20q^-34 - 5q^-33 - 14q^-32 - 4q^-31 + 2q^-30 + 17q^-29 - 2q^-28 - 8q^-27 - 4q^-26 - 3q^-25 + 9q^-24 + q^-22 - q^-21 - 4q^-20 + 2q^-19 - q^-18 + 2q^-17 + q^-16 - q^-13 + q^-8

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]:=
PD[Knot[10, 145]]

Out[2]=
PD[X[4, 2, 5, 1], X[5, 12, 6, 13], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[11, 16, 12, 17], X[17, 10, 18, 11], X[7, 18, 8, 19], X[13, 20, 14, 1], > X[19, 14, 20, 15], X[15, 6, 16, 7]]

In[3]:=
GaussCode[Knot[10, 145]]

Out[3]=
GaussCode[1, -4, 3, -1, -2, 10, -7, -3, 4, 6, -5, 2, -8, 9, -10, 5, -6, 7, -9, > 8]

In[4]:=
DTCode[Knot[10, 145]]

Out[4]=
DTCode[4, 8, -12, -18, 2, -16, -20, -6, -10, -14]

In[5]:=
br = BR[Knot[10, 145]]

Out[5]=
BR[4, {-1, -1, -2, 1, -2, -1, -3, -2, 1, -2, -3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 145]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 145]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 145]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 2, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 145]][t]

Out[10]=
-2 1 2 -3 + t + - + t + t t

In[11]:=
Conway[Knot[10, 145]][z]

Out[11]=
2 4 1 + 5 z + z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 145]}

In[13]:=
{KnotDet[Knot[10, 145]], KnotSignature[Knot[10, 145]]}

Out[13]=
{3, -2}

In[14]:=
Jones[Knot[10, 145]][q]

Out[14]=
-10 -9 -8 -7 -2 -q + q - q + q + q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 145]}

In[16]:=
A2Invariant[Knot[10, 145]][q]

Out[16]=
-32 -30 -24 -14 -10 -8 -6 -q - q + q + q + q + q + q

In[17]:=
HOMFLYPT[Knot[10, 145]][a, z]

Out[17]=
4 6 8 10 4 2 8 2 4 4 2 a - a + a - a + 4 a z + a z + a z

In[18]:=
Kauffman[Knot[10, 145]][a, z]

Out[18]=
4 6 8 10 5 7 9 11 4 2 6 2 2 a + a + a + a - a z - 2 a z - 6 a z - 5 a z - 4 a z - 2 a z - 8 2 10 2 7 3 9 3 11 3 4 4 8 4 > 4 a z - 6 a z + 8 a z + 18 a z + 10 a z + a z + 9 a z + 10 4 7 5 9 5 11 5 8 6 10 6 7 7 > 10 a z - 6 a z - 12 a z - 6 a z - 6 a z - 6 a z + a z + 9 7 11 7 8 8 10 8 > 2 a z + a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 145]], Vassiliev[3][Knot[10, 145]]}

Out[19]=
{5, -12}

In[20]:=
Kh[Knot[10, 145]][q, t]

Out[20]=
-5 -3 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 9 17 8 17 7 15 6 13 6 15 5 11 5 q t q t q t q t q t q t q t 2 1 1 1 > ------ + ------ + ----- + ----- 11 4 11 3 7 3 7 2 q t q t q t q t

In[21]:=
ColouredJones[Knot[10, 145], 2][q]

Out[21]=
-29 -28 -27 2 -25 -24 2 2 -21 2 4 2 q - q - q + --- - q - q + --- - --- + q + --- - --- + --- + 26 23 22 20 19 18 q q q q q q 2 5 2 2 5 4 2 4 2 -8 -7 -4 > --- - --- + --- + --- - --- + --- + --- - --- + -- + q - q + q 17 16 15 14 13 12 11 10 9 q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₄₅

10₁₄₄
10₁₄₆

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-29 - q^-28 - q^-27 + 2q^-26 - q^-25 - q^-24 + 2q^-23 - 2q^-22 + q^-21 + 2q^-20 - 4q^-19 + 2q^-18 + 2q^-17 - 5q^-16 + 2q^-15 + 2q^-14 - 5q^-13 + 4q^-12 + 2q^-11 - 4q^-10 + 2q^-9 + q^-8 - q^-7 + q^-4
3	- q^-57 + q^-56 + q^-55 - 2q^-53 + 2q^-51 - 2q^-49 + 3q^-47 - 2q^-46 - 3q^-45 + 2q^-44 + 5q^-43 - 2q^-42 - 6q^-41 + 2q^-40 + 6q^-39 - q^-38 - 5q^-37 + 4q^-35 - 3q^-33 + q^-31 + q^-30 - q^-29 - q^-28 + q^-26 + q^-25 - 2q^-24 - q^-23 + q^-22 + 2q^-21 - 2q^-20 - 2q^-19 + q^-18 + 3q^-17 - 2q^-15 - q^-14 + 2q^-13 + q^-12 - q^-10 + q^-6
4	q^-94 - q^-93 - q^-92 + 3q^-89 - q^-88 - q^-87 - q^-86 - q^-85 + 4q^-84 - q^-83 - q^-82 - 2q^-81 + q^-80 + 5q^-79 - 2q^-78 - 3q^-77 - 5q^-76 + 3q^-75 + 7q^-74 + q^-73 - 3q^-72 - 8q^-71 + 2q^-70 + 6q^-69 + q^-68 - 2q^-67 - 4q^-66 + 3q^-65 + 4q^-64 - 3q^-63 - 5q^-62 - q^-61 + 6q^-60 + 8q^-59 - 6q^-58 - 10q^-57 - q^-56 + 7q^-55 + 13q^-54 - 6q^-53 - 12q^-52 - 3q^-51 + 5q^-50 + 18q^-49 - 5q^-48 - 14q^-47 - 5q^-46 + 4q^-45 + 21q^-44 - 5q^-43 - 16q^-42 - 4q^-41 + 3q^-40 + 21q^-39 - 6q^-38 - 16q^-37 - 2q^-36 + 3q^-35 + 20q^-34 - 5q^-33 - 14q^-32 - 4q^-31 + 2q^-30 + 17q^-29 - 2q^-28 - 8q^-27 - 4q^-26 - 3q^-25 + 9q^-24 + q^-22 - q^-21 - 4q^-20 + 2q^-19 - q^-18 + 2q^-17 + q^-16 - q^-13 + q^-8

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 145]]
Out[2]=	PD[X[4, 2, 5, 1], X[5, 12, 6, 13], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[11, 16, 12, 17], X[17, 10, 18, 11], X[7, 18, 8, 19], X[13, 20, 14, 1], > X[19, 14, 20, 15], X[15, 6, 16, 7]]
In[3]:=	GaussCode[Knot[10, 145]]
Out[3]=	GaussCode[1, -4, 3, -1, -2, 10, -7, -3, 4, 6, -5, 2, -8, 9, -10, 5, -6, 7, -9, > 8]
In[4]:=	DTCode[Knot[10, 145]]
Out[4]=	DTCode[4, 8, -12, -18, 2, -16, -20, -6, -10, -14]
In[5]:=	br = BR[Knot[10, 145]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -1, -3, -2, 1, -2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 145]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 145]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 145]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 145]][t]
Out[10]=	-2 1 2 -3 + t + - + t + t t
In[11]:=	Conway[Knot[10, 145]][z]
Out[11]=	2 4 1 + 5 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 145]}
In[13]:=	{KnotDet[Knot[10, 145]], KnotSignature[Knot[10, 145]]}
Out[13]=	{3, -2}
In[14]:=	Jones[Knot[10, 145]][q]
Out[14]=	-10 -9 -8 -7 -2 -q + q - q + q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 145]}
In[16]:=	A2Invariant[Knot[10, 145]][q]
Out[16]=	-32 -30 -24 -14 -10 -8 -6 -q - q + q + q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 145]][a, z]
Out[17]=	4 6 8 10 4 2 8 2 4 4 2 a - a + a - a + 4 a z + a z + a z
In[18]:=	Kauffman[Knot[10, 145]][a, z]
Out[18]=	4 6 8 10 5 7 9 11 4 2 6 2 2 a + a + a + a - a z - 2 a z - 6 a z - 5 a z - 4 a z - 2 a z - 8 2 10 2 7 3 9 3 11 3 4 4 8 4 > 4 a z - 6 a z + 8 a z + 18 a z + 10 a z + a z + 9 a z + 10 4 7 5 9 5 11 5 8 6 10 6 7 7 > 10 a z - 6 a z - 12 a z - 6 a z - 6 a z - 6 a z + a z + 9 7 11 7 8 8 10 8 > 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 145]], Vassiliev[3][Knot[10, 145]]}
Out[19]=	{5, -12}
In[20]:=	Kh[Knot[10, 145]][q, t]
Out[20]=	-5 -3 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 21 9 17 8 17 7 15 6 13 6 15 5 11 5 q t q t q t q t q t q t q t 2 1 1 1 > ------ + ------ + ----- + ----- 11 4 11 3 7 3 7 2 q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 145], 2][q]
Out[21]=	-29 -28 -27 2 -25 -24 2 2 -21 2 4 2 q - q - q + --- - q - q + --- - --- + q + --- - --- + --- + 26 23 22 20 19 18 q q q q q q 2 5 2 2 5 4 2 4 2 -8 -7 -4 > --- - --- + --- + --- - --- + --- + --- - --- + -- + q - q + q 17 16 15 14 13 12 11 10 9 q q q q q q q q q

The Non Alternating Knot 10145

The Non Alternating Knot 10₁₄₅