The Alternating Knot 10₁₄

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_5,12,6,13 X_3,11,4,10 X_11,3,12,2 X_7,16,8,17 X_13,20,14,1 X_19,14,20,15 X_9,18,10,19 X_15,6,16,7 X_17,8,18,9

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

DT (Dowker-Thistlethwaite) Code: 4 10 12 16 18 2 20 6 8 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 3 2 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 8t^-2 - 12t^-1 + 13 - 12t + 8t² - 2t³

Conway Polynomial: 1 + 2z² - 4z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {K11a161, K11n2, ...}

Determinant and Signature: {57, -4}

Jones Polynomial: q^-10 - 3q^-9 + 5q^-8 - 8q^-7 + 9q^-6 - 9q^-5 + 9q^-4 - 6q^-3 + 4q^-2 - 2q^-1 + 1

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

A2 (sl(3)) Invariant: q^-30 - q^-28 - 2q^-22 + q^-20 - 2q^-18 + q^-16 + q^-14 + 3q^-10 - q^-8 + q^-6 + 1

HOMFLY-PT Polynomial: a² + 3a²z² + a²z⁴ + a⁴ - a⁴z² - 3a⁴z⁴ - a⁴z⁶ - a⁶ - 2a⁶z² - 3a⁶z⁴ - a⁶z⁶ + 2a⁸z² + a⁸z⁴

Kauffman Polynomial: - a² + 4a²z² - 4a²z⁴ + a²z⁶ - a³z + 6a³z³ - 7a³z⁵ + 2a³z⁷ + a⁴ - a⁴z² + 2a⁴z⁴ - 5a⁴z⁶ + 2a⁴z⁸ - 4a⁵z + 10a⁵z³ - 9a⁵z⁵ + a⁵z⁷ + a⁵z⁹ + a⁶ - 9a⁶z² + 16a⁶z⁴ - 14a⁶z⁶ + 5a⁶z⁸ - 2a⁷z + 8a⁷z³ - 9a⁷z⁵ + 3a⁷z⁷ + a⁷z⁹ - 3a⁸z² + 5a⁸z⁴ - 4a⁸z⁶ + 3a⁸z⁸ + 2a⁹z - 4a⁹z⁵ + 4a⁹z⁷ - 4a¹⁰z⁴ + 4a¹⁰z⁶ + a¹¹z - 4a¹¹z³ + 3a¹¹z⁵ - a¹²z² + a¹²z⁴

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

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

t^rq^j 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 1

j = -3 3 1

j = -5 4 2

j = -7 5 2

j = -9 4 4

j = -11 5 5

j = -13 3 4

j = -15 2 5

j = -17 1 3

j = -19 2

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-28 - 3q^-27 + q^-26 + 7q^-25 - 13q^-24 + 4q^-23 + 21q^-22 - 33q^-21 + 5q^-20 + 44q^-19 - 55q^-18 + q^-17 + 64q^-16 - 65q^-15 - 7q^-14 + 71q^-13 - 56q^-12 - 16q^-11 + 62q^-10 - 36q^-9 - 20q^-8 + 41q^-7 - 15q^-6 - 16q^-5 + 20q^-4 - 3q^-3 - 8q^-2 + 6q^-1 - 2q + q²

3 q^-54 - 3q^-53 + q^-52 + 3q^-51 + 2q^-50 - 8q^-49 - 2q^-48 + 14q^-47 - 21q^-45 + 37q^-43 - 5q^-42 - 57q^-41 + 3q^-40 + 92q^-39 - 6q^-38 - 122q^-37 - 9q^-36 + 168q^-35 + 18q^-34 - 197q^-33 - 43q^-32 + 227q^-31 + 64q^-30 - 240q^-29 - 87q^-28 + 237q^-27 + 113q^-26 - 231q^-25 - 126q^-24 + 201q^-23 + 148q^-22 - 178q^-21 - 149q^-20 + 132q^-19 + 163q^-18 - 103q^-17 - 149q^-16 + 56q^-15 + 142q^-14 - 27q^-13 - 116q^-12 - 3q^-11 + 94q^-10 + 16q^-9 - 63q^-8 - 26q^-7 + 43q^-6 + 21q^-5 - 21q^-4 - 19q^-3 + 13q^-2 + 10q^-1 - 4 - 7q + 3q² + 2q³ - 2q⁵ + q⁶

4 q^-88 - 3q^-87 + q^-86 + 3q^-85 - 2q^-84 + 7q^-83 - 14q^-82 + 2q^-81 + 8q^-80 - 9q^-79 + 28q^-78 - 34q^-77 + 8q^-76 + 14q^-75 - 40q^-74 + 57q^-73 - 53q^-72 + 48q^-71 + 38q^-70 - 116q^-69 + 48q^-68 - 100q^-67 + 160q^-66 + 149q^-65 - 213q^-64 - 57q^-63 - 247q^-62 + 323q^-61 + 398q^-60 - 239q^-59 - 226q^-58 - 545q^-57 + 427q^-56 + 730q^-55 - 127q^-54 - 345q^-53 - 908q^-52 + 395q^-51 + 984q^-50 + 70q^-49 - 325q^-48 - 1178q^-47 + 254q^-46 + 1055q^-45 + 245q^-44 - 188q^-43 - 1272q^-42 + 82q^-41 + 951q^-40 + 356q^-39 + 9q^-38 - 1210q^-37 - 92q^-36 + 730q^-35 + 412q^-34 + 230q^-33 - 1020q^-32 - 259q^-31 + 424q^-30 + 404q^-29 + 442q^-28 - 723q^-27 - 360q^-26 + 96q^-25 + 292q^-24 + 554q^-23 - 369q^-22 - 319q^-21 - 148q^-20 + 98q^-19 + 501q^-18 - 82q^-17 - 165q^-16 - 217q^-15 - 66q^-14 + 319q^-13 + 45q^-12 - 15q^-11 - 146q^-10 - 115q^-9 + 139q^-8 + 42q^-7 + 42q^-6 - 54q^-5 - 77q^-4 + 43q^-3 + 10q^-2 + 30q^-1 - 10 - 31q + 13q² - 2q³ + 10q⁴ - 9q⁶ + 4q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

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, 14]]

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

In[3]:=
GaussCode[Knot[10, 14]]

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

In[4]:=
DTCode[Knot[10, 14]]

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

In[5]:=
br = BR[Knot[10, 14]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 14]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 14]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 14]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 14]][t]

Out[10]=
2 8 12 2 3 13 - -- + -- - -- - 12 t + 8 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 14]][z]

Out[11]=
2 4 6 1 + 2 z - 4 z - 2 z

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

Out[12]=
{Knot[10, 14], Knot[11, Alternating, 161], Knot[11, NonAlternating, 2]}

In[13]:=
{KnotDet[Knot[10, 14]], KnotSignature[Knot[10, 14]]}

Out[13]=
{57, -4}

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

Out[14]=
-10 3 5 8 9 9 9 6 4 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q

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

Out[15]=
{Knot[10, 14]}

In[16]:=
A2Invariant[Knot[10, 14]][q]

Out[16]=
-30 -28 2 -20 2 -16 -14 3 -8 -6 1 + q - q - --- + q - --- + q + q + --- - q + q 22 18 10 q q q

In[17]:=
HOMFLYPT[Knot[10, 14]][a, z]

Out[17]=
2 4 6 2 2 4 2 6 2 8 2 2 4 4 4 a + a - a + 3 a z - a z - 2 a z + 2 a z + a z - 3 a z - 6 4 8 4 4 6 6 6 > 3 a z + a z - a z - a z

In[18]:=
Kauffman[Knot[10, 14]][a, z]

Out[18]=
2 4 6 3 5 7 9 11 2 2 4 2 -a + a + a - a z - 4 a z - 2 a z + 2 a z + a z + 4 a z - a z - 6 2 8 2 12 2 3 3 5 3 7 3 11 3 > 9 a z - 3 a z - a z + 6 a z + 10 a z + 8 a z - 4 a z - 2 4 4 4 6 4 8 4 10 4 12 4 3 5 > 4 a z + 2 a z + 16 a z + 5 a z - 4 a z + a z - 7 a z - 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 9 a z - 9 a z - 4 a z + 3 a z + a z - 5 a z - 14 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 4 a z + 4 a z + 2 a z + a z + 3 a z + 4 a z + 2 a z + 6 8 8 8 5 9 7 9 > 5 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 14]], Vassiliev[3][Knot[10, 14]]}

Out[19]=
{2, -3}

In[20]:=
Kh[Knot[10, 14]][q, t]

Out[20]=
2 3 1 2 1 3 2 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 4 5 5 4 4 5 2 4 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 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 t q t q 2 > q t

In[21]:=
ColouredJones[Knot[10, 14], 2][q]

Out[21]=
-28 3 -26 7 13 4 21 33 5 44 55 -17 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + q + 27 25 24 23 22 21 20 19 18 q q q q q q q q q 64 65 7 71 56 16 62 36 20 41 15 16 20 > --- - --- - --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q q q 3 8 6 2 > -- - -- + - - 2 q + q 3 2 q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₄

10₁₃
10₁₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 3q^-27 + q^-26 + 7q^-25 - 13q^-24 + 4q^-23 + 21q^-22 - 33q^-21 + 5q^-20 + 44q^-19 - 55q^-18 + q^-17 + 64q^-16 - 65q^-15 - 7q^-14 + 71q^-13 - 56q^-12 - 16q^-11 + 62q^-10 - 36q^-9 - 20q^-8 + 41q^-7 - 15q^-6 - 16q^-5 + 20q^-4 - 3q^-3 - 8q^-2 + 6q^-1 - 2q + q²
3	q^-54 - 3q^-53 + q^-52 + 3q^-51 + 2q^-50 - 8q^-49 - 2q^-48 + 14q^-47 - 21q^-45 + 37q^-43 - 5q^-42 - 57q^-41 + 3q^-40 + 92q^-39 - 6q^-38 - 122q^-37 - 9q^-36 + 168q^-35 + 18q^-34 - 197q^-33 - 43q^-32 + 227q^-31 + 64q^-30 - 240q^-29 - 87q^-28 + 237q^-27 + 113q^-26 - 231q^-25 - 126q^-24 + 201q^-23 + 148q^-22 - 178q^-21 - 149q^-20 + 132q^-19 + 163q^-18 - 103q^-17 - 149q^-16 + 56q^-15 + 142q^-14 - 27q^-13 - 116q^-12 - 3q^-11 + 94q^-10 + 16q^-9 - 63q^-8 - 26q^-7 + 43q^-6 + 21q^-5 - 21q^-4 - 19q^-3 + 13q^-2 + 10q^-1 - 4 - 7q + 3q² + 2q³ - 2q⁵ + q⁶
4	q^-88 - 3q^-87 + q^-86 + 3q^-85 - 2q^-84 + 7q^-83 - 14q^-82 + 2q^-81 + 8q^-80 - 9q^-79 + 28q^-78 - 34q^-77 + 8q^-76 + 14q^-75 - 40q^-74 + 57q^-73 - 53q^-72 + 48q^-71 + 38q^-70 - 116q^-69 + 48q^-68 - 100q^-67 + 160q^-66 + 149q^-65 - 213q^-64 - 57q^-63 - 247q^-62 + 323q^-61 + 398q^-60 - 239q^-59 - 226q^-58 - 545q^-57 + 427q^-56 + 730q^-55 - 127q^-54 - 345q^-53 - 908q^-52 + 395q^-51 + 984q^-50 + 70q^-49 - 325q^-48 - 1178q^-47 + 254q^-46 + 1055q^-45 + 245q^-44 - 188q^-43 - 1272q^-42 + 82q^-41 + 951q^-40 + 356q^-39 + 9q^-38 - 1210q^-37 - 92q^-36 + 730q^-35 + 412q^-34 + 230q^-33 - 1020q^-32 - 259q^-31 + 424q^-30 + 404q^-29 + 442q^-28 - 723q^-27 - 360q^-26 + 96q^-25 + 292q^-24 + 554q^-23 - 369q^-22 - 319q^-21 - 148q^-20 + 98q^-19 + 501q^-18 - 82q^-17 - 165q^-16 - 217q^-15 - 66q^-14 + 319q^-13 + 45q^-12 - 15q^-11 - 146q^-10 - 115q^-9 + 139q^-8 + 42q^-7 + 42q^-6 - 54q^-5 - 77q^-4 + 43q^-3 + 10q^-2 + 30q^-1 - 10 - 31q + 13q² - 2q³ + 10q⁴ - 9q⁶ + 4q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 14]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 14]][t]
Out[10]=	2 8 12 2 3 13 - -- + -- - -- - 12 t + 8 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 14]][z]
Out[11]=	2 4 6 1 + 2 z - 4 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 14], Knot[11, Alternating, 161], Knot[11, NonAlternating, 2]}
In[13]:=	{KnotDet[Knot[10, 14]], KnotSignature[Knot[10, 14]]}
Out[13]=	{57, -4}
In[14]:=	Jones[Knot[10, 14]][q]
Out[14]=	-10 3 5 8 9 9 9 6 4 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 14]}
In[16]:=	A2Invariant[Knot[10, 14]][q]
Out[16]=	-30 -28 2 -20 2 -16 -14 3 -8 -6 1 + q - q - --- + q - --- + q + q + --- - q + q 22 18 10 q q q
In[17]:=	HOMFLYPT[Knot[10, 14]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 8 2 2 4 4 4 a + a - a + 3 a z - a z - 2 a z + 2 a z + a z - 3 a z - 6 4 8 4 4 6 6 6 > 3 a z + a z - a z - a z
In[18]:=	Kauffman[Knot[10, 14]][a, z]
Out[18]=	2 4 6 3 5 7 9 11 2 2 4 2 -a + a + a - a z - 4 a z - 2 a z + 2 a z + a z + 4 a z - a z - 6 2 8 2 12 2 3 3 5 3 7 3 11 3 > 9 a z - 3 a z - a z + 6 a z + 10 a z + 8 a z - 4 a z - 2 4 4 4 6 4 8 4 10 4 12 4 3 5 > 4 a z + 2 a z + 16 a z + 5 a z - 4 a z + a z - 7 a z - 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 9 a z - 9 a z - 4 a z + 3 a z + a z - 5 a z - 14 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 4 a z + 4 a z + 2 a z + a z + 3 a z + 4 a z + 2 a z + 6 8 8 8 5 9 7 9 > 5 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 14]], Vassiliev[3][Knot[10, 14]]}
Out[19]=	{2, -3}
In[20]:=	Kh[Knot[10, 14]][q, t]
Out[20]=	2 3 1 2 1 3 2 5 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 4 5 5 4 4 5 2 4 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 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 t q t q 2 > q t
In[21]:=	ColouredJones[Knot[10, 14], 2][q]
Out[21]=	-28 3 -26 7 13 4 21 33 5 44 55 -17 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + q + 27 25 24 23 22 21 20 19 18 q q q q q q q q q 64 65 7 71 56 16 62 36 20 41 15 16 20 > --- - --- - --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q q q 3 8 6 2 > -- - -- + - - 2 q + q 3 2 q q q

The Alternating Knot 1014

The Alternating Knot 10₁₄