8 14

8_13

8_15

Visit 8 14's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 14's page at Knotilus! Visit 8 14's page at the original Knot Atlas! 8 14 Quick Notes

8 14 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,10,6,11 X3948 X9,3,10,2 X7,14,8,15 X11,16,12,1 X15,12,16,13 X13,6,14,7
Gauss code	-1, 4, -3, 1, -2, 8, -5, 3, -4, 2, -6, 7, -8, 5, -7, 6
Dowker-Thistlethwaite code	4 8 10 14 2 16 6 12
Conway Notation	[22112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-9][-1] Hyperbolic Volume 9.2178 A-Polynomial See Data:8 14/A-polynomial

[edit Notes for 8 14's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant -2

[edit Notes for 8 14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+8t-11+8t^{-1}-2t^{-2}$
Conway polynomial	$1-2z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 31, -2 }
Jones polynomial	$q-2+4q^{-1}-5q^{-2}+6q^{-3}-5q^{-4}+4q^{-5}-3q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}-z^{4}a^{4}-z^{2}a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-5z^{3}a^{7}+za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+z^{2}a^{6}+z^{7}a^{5}+4z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+3z^{2}a^{4}+z^{7}a^{3}+3z^{5}a^{3}-6z^{3}a^{3}+3za^{3}+2z^{6}a^{2}-z^{4}a^{2}-z^{2}a^{2}+2z^{5}a-3z^{3}a+za+z^{4}-2z^{2}+1$
The A2 invariant	$q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^{6}-q^{4}+2q^{2}+q^{-4}$
The G2 invariant	$q^{114}-2q^{112}+4q^{110}-6q^{108}+3q^{106}-6q^{102}+14q^{100}-16q^{98}+17q^{96}-11q^{94}-4q^{92}+17q^{90}-25q^{88}+25q^{86}-17q^{84}+4q^{82}+11q^{80}-17q^{78}+17q^{76}-9q^{74}-3q^{72}+13q^{70}-16q^{68}+5q^{66}+7q^{64}-19q^{62}+28q^{60}-26q^{58}+15q^{56}+q^{54}-21q^{52}+33q^{50}-37q^{48}+28q^{46}-11q^{44}-7q^{42}+21q^{40}-23q^{38}+19q^{36}-7q^{34}-6q^{32}+13q^{30}-12q^{28}+2q^{26}+12q^{24}-19q^{22}+22q^{20}-13q^{18}+12q^{14}-21q^{12}+24q^{10}-18q^{8}+8q^{6}+2q^{4}-9q^{2}+12-10q^{-2}+9q^{-4}-3q^{-6}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

Further Quantum Invariants

Further quantum knot invariants for 8_14.

A1 Invariants.

Weight	Invariant
1	$q^{15}-2q^{13}+q^{11}-q^{9}+q^{7}+q^{5}-q^{3}+2q-q^{-1}+q^{-3}$
2	$q^{42}-2q^{40}-2q^{38}+6q^{36}-q^{34}-6q^{32}+7q^{30}+q^{28}-8q^{26}+4q^{24}+2q^{22}-5q^{20}+3q^{16}+2q^{14}-5q^{12}+2q^{10}+7q^{8}-7q^{6}-q^{4}+8q^{2}-4-2q^{-2}+4q^{-4}-q^{-6}-q^{-8}+q^{-10}$
3	$q^{81}-2q^{79}-2q^{77}+3q^{75}+6q^{73}-q^{71}-13q^{69}-q^{67}+16q^{65}+7q^{63}-19q^{61}-15q^{59}+19q^{57}+22q^{55}-16q^{53}-25q^{51}+12q^{49}+26q^{47}-5q^{45}-25q^{43}+2q^{41}+19q^{39}+3q^{37}-15q^{35}-8q^{33}+6q^{31}+13q^{29}-18q^{25}-7q^{23}+20q^{21}+16q^{19}-20q^{17}-21q^{15}+19q^{13}+25q^{11}-12q^{9}-25q^{7}+6q^{5}+23q^{3}-q-16q^{-1}-2q^{-3}+11q^{-5}+3q^{-7}-6q^{-9}-2q^{-11}+3q^{-13}+q^{-15}-q^{-17}-q^{-19}+q^{-21}$
4	$q^{132}-2q^{130}-2q^{128}+3q^{126}+3q^{124}+6q^{122}-8q^{120}-13q^{118}-q^{116}+8q^{114}+31q^{112}-2q^{110}-33q^{108}-27q^{106}-2q^{104}+65q^{102}+34q^{100}-30q^{98}-68q^{96}-50q^{94}+74q^{92}+84q^{90}+9q^{88}-84q^{86}-101q^{84}+43q^{82}+102q^{80}+57q^{78}-60q^{76}-116q^{74}+3q^{72}+81q^{70}+73q^{68}-23q^{66}-91q^{64}-24q^{62}+41q^{60}+62q^{58}+8q^{56}-49q^{54}-42q^{52}-q^{50}+45q^{48}+42q^{46}-q^{44}-65q^{42}-46q^{40}+28q^{38}+75q^{36}+49q^{34}-74q^{32}-88q^{30}-6q^{28}+90q^{26}+99q^{24}-53q^{22}-104q^{20}-48q^{18}+64q^{16}+115q^{14}-7q^{12}-73q^{10}-68q^{8}+14q^{6}+85q^{4}+22q^{2}-24-49q^{-2}-15q^{-4}+37q^{-6}+17q^{-8}+2q^{-10}-19q^{-12}-12q^{-14}+11q^{-16}+4q^{-18}+4q^{-20}-4q^{-22}-4q^{-24}+3q^{-26}+q^{-30}-q^{-32}-q^{-34}+q^{-36}$
5	$q^{195}-2q^{193}-2q^{191}+3q^{189}+3q^{187}+3q^{185}-q^{183}-8q^{181}-13q^{179}-q^{177}+17q^{175}+23q^{173}+13q^{171}-16q^{169}-43q^{167}-44q^{165}+10q^{163}+70q^{161}+78q^{159}+23q^{157}-76q^{155}-138q^{153}-86q^{151}+67q^{149}+189q^{147}+165q^{145}-8q^{143}-220q^{141}-266q^{139}-76q^{137}+215q^{135}+352q^{133}+184q^{131}-165q^{129}-404q^{127}-295q^{125}+85q^{123}+414q^{121}+381q^{119}+7q^{117}-379q^{115}-432q^{113}-94q^{111}+316q^{109}+431q^{107}+164q^{105}-241q^{103}-402q^{101}-194q^{99}+161q^{97}+341q^{95}+211q^{93}-89q^{91}-279q^{89}-199q^{87}+30q^{85}+203q^{83}+196q^{81}+28q^{79}-146q^{77}-184q^{75}-83q^{73}+84q^{71}+187q^{69}+147q^{67}-27q^{65}-200q^{63}-214q^{61}-37q^{59}+202q^{57}+287q^{55}+114q^{53}-199q^{51}-361q^{49}-193q^{47}+170q^{45}+411q^{43}+284q^{41}-114q^{39}-432q^{37}-367q^{35}+39q^{33}+407q^{31}+416q^{29}+57q^{27}-339q^{25}-427q^{23}-145q^{21}+245q^{19}+392q^{17}+203q^{15}-130q^{13}-316q^{11}-226q^{9}+33q^{7}+228q^{5}+205q^{3}+31q-132q^{-1}-161q^{-3}-62q^{-5}+63q^{-7}+109q^{-9}+60q^{-11}-19q^{-13}-61q^{-15}-44q^{-17}-q^{-19}+29q^{-21}+28q^{-23}+5q^{-25}-13q^{-27}-13q^{-29}-3q^{-31}+2q^{-33}+6q^{-35}+4q^{-37}-3q^{-39}-2q^{-41}+q^{-43}+q^{-49}-q^{-51}-q^{-53}+q^{-55}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-q^{20}-q^{18}+q^{16}-q^{14}+q^{12}+q^{6}-q^{4}+2q^{2}+q^{-4}$
1,1	$q^{60}-4q^{58}+10q^{56}-20q^{54}+32q^{52}-48q^{50}+66q^{48}-78q^{46}+83q^{44}-78q^{42}+64q^{40}-36q^{38}-q^{36}+42q^{34}-84q^{32}+116q^{30}-145q^{28}+156q^{26}-154q^{24}+138q^{22}-105q^{20}+68q^{18}-26q^{16}-12q^{14}+47q^{12}-68q^{10}+78q^{8}-76q^{6}+70q^{4}-56q^{2}+42-28q^{-2}+19q^{-4}-10q^{-6}+6q^{-8}-2q^{-10}+q^{-12}$
2,0	$q^{56}-q^{54}-2q^{52}+3q^{48}+2q^{46}-4q^{44}+4q^{40}+q^{38}-4q^{36}-q^{34}+3q^{32}-2q^{30}-4q^{28}+q^{24}-2q^{22}+3q^{20}+3q^{18}-q^{16}+q^{14}+4q^{12}-5q^{8}+5q^{4}-q^{2}-3+2q^{-2}+3q^{-4}-q^{-8}+q^{-12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-2q^{46}+3q^{42}-5q^{40}+2q^{38}+5q^{36}-6q^{34}+q^{32}+5q^{30}-4q^{28}-q^{26}+3q^{24}-q^{22}-2q^{20}-2q^{18}+3q^{16}-q^{14}-4q^{12}+7q^{10}+q^{8}-5q^{6}+6q^{4}+q^{2}-3+3q^{-2}+q^{-4}-q^{-6}+q^{-8}$
1,0,0	$q^{29}-q^{27}-q^{23}+q^{21}-q^{19}+q^{17}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-5}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}-q^{60}-2q^{58}+2q^{56}+2q^{54}-3q^{52}-2q^{50}+4q^{48}+q^{46}-5q^{44}-q^{42}+6q^{40}-3q^{36}+5q^{34}+3q^{32}-4q^{30}-q^{28}+q^{26}-5q^{24}-5q^{22}+2q^{20}+2q^{18}-4q^{16}+2q^{14}+8q^{12}+q^{10}-3q^{8}+3q^{6}+3q^{4}-q^{2}-1+2q^{-2}+2q^{-4}+q^{-10}$
1,0,0,0	$q^{36}-q^{34}-q^{28}+q^{26}-q^{24}+q^{22}+q^{8}-q^{6}+2q^{4}+1+q^{-2}+q^{-6}$

B2 Invariants.

Weight	Invariant
0,1	$q^{48}-2q^{46}+4q^{44}-5q^{42}+5q^{40}-6q^{38}+5q^{36}-4q^{34}+q^{32}+q^{30}-4q^{28}+7q^{26}-9q^{24}+11q^{22}-10q^{20}+10q^{18}-7q^{16}+5q^{14}-2q^{12}-q^{10}+3q^{8}-5q^{6}+6q^{4}-5q^{2}+5-3q^{-2}+3q^{-4}-q^{-6}+q^{-8}$
1,0	$q^{78}-2q^{74}-2q^{72}+2q^{70}+4q^{68}-q^{66}-5q^{64}-2q^{62}+6q^{60}+5q^{58}-3q^{56}-6q^{54}+5q^{50}+2q^{48}-4q^{46}-3q^{44}+3q^{42}+3q^{40}-2q^{38}-4q^{36}+q^{34}+4q^{32}-5q^{28}-q^{26}+4q^{24}+2q^{22}-4q^{20}-3q^{18}+4q^{16}+6q^{14}-q^{12}-6q^{10}-q^{8}+6q^{6}+4q^{4}-2q^{2}-4+3q^{-4}+2q^{-6}-q^{-8}-q^{-10}+q^{-14}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{66}-2q^{64}+2q^{62}-3q^{60}+4q^{58}-5q^{56}+4q^{54}-4q^{52}+5q^{50}-3q^{48}+q^{46}+q^{42}+3q^{40}-5q^{38}+5q^{36}-6q^{34}+8q^{32}-9q^{30}+6q^{28}-9q^{26}+7q^{24}-5q^{22}+3q^{20}-3q^{18}+2q^{16}+3q^{14}-q^{12}+3q^{10}-4q^{8}+6q^{6}-3q^{4}+4q^{2}-3+4q^{-2}-q^{-4}+2q^{-6}-q^{-8}+q^{-10}$

G2 Invariants.

Weight

Invariant

1,0

$q^{114}-2q^{112}+4q^{110}-6q^{108}+3q^{106}-6q^{102}+14q^{100}-16q^{98}+17q^{96}-11q^{94}-4q^{92}+17q^{90}-25q^{88}+25q^{86}-17q^{84}+4q^{82}+11q^{80}-17q^{78}+17q^{76}-9q^{74}-3q^{72}+13q^{70}-16q^{68}+5q^{66}+7q^{64}-19q^{62}+28q^{60}-26q^{58}+15q^{56}+q^{54}-21q^{52}+33q^{50}-37q^{48}+28q^{46}-11q^{44}-7q^{42}+21q^{40}-23q^{38}+19q^{36}-7q^{34}-6q^{32}+13q^{30}-12q^{28}+2q^{26}+12q^{24}-19q^{22}+22q^{20}-13q^{18}+12q^{14}-21q^{12}+24q^{10}-18q^{8}+8q^{6}+2q^{4}-9q^{2}+12-10q^{-2}+9q^{-4}-3q^{-6}+2q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 14"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-2t^{2}+8t-11+8t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-2z^{4}$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 31, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q-2+4q^{-1}-5q^{-2}+6q^{-3}-5q^{-4}+4q^{-5}-3q^{-6}+q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{2}a^{6}-z^{4}a^{4}-z^{2}a^{4}-z^{4}a^{2}-z^{2}a^{2}+z^{2}+1$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-5z^{3}a^{7}+za^{7}+3z^{6}a^{6}-4z^{4}a^{6}+z^{2}a^{6}+z^{7}a^{5}+4z^{5}a^{5}-8z^{3}a^{5}+3za^{5}+5z^{6}a^{4}-7z^{4}a^{4}+3z^{2}a^{4}+z^{7}a^{3}+3z^{5}a^{3}-6z^{3}a^{3}+3za^{3}+2z^{6}a^{2}-z^{4}a^{2}-z^{2}a^{2}+2z^{5}a-3z^{3}a+za+z^{4}-2z^{2}+1$

Vassiliev invariants

V2 and V3:

(0, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $0$ $0$ $16$ $16$ $0$ $-64$ $-32$ $-32$ $0$ $0$ $0$ $0$ $136$ ${\frac {32}{3}}$ ${\frac {320}{3}}$ ${\frac {56}{3}}$ $8$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 8 14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 3                 1 1 1               1   -1 -1             3 1   2 -3           3 2     -1 -5         3 2       1 -7       2 3         1 -9     2 3           -1 -11   1 2             1 -13   2               -2 -15 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[8, 14]]
Out[2]=	8
In[3]:=	PD[Knot[8, 14]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[7, 14, 8, 15], X[11, 16, 12, 1], X[15, 12, 16, 13], X[13, 6, 14, 7]]
In[4]:=	GaussCode[Knot[8, 14]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 8, -5, 3, -4, 2, -6, 7, -8, 5, -7, 6]
In[5]:=	BR[Knot[8, 14]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[8, 14]][t]
Out[6]=	2 8 2 -11 - -- + - + 8 t - 2 t 2 t t
In[7]:=	Conway[Knot[8, 14]][z]
Out[7]=	4 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 14], Knot[9, 8], Knot[10, 131]}
In[9]:=	{KnotDet[Knot[8, 14]], KnotSignature[Knot[8, 14]]}
Out[9]=	{31, -2}
In[10]:=	J=Jones[Knot[8, 14]][q]
Out[10]=	-7 3 4 5 6 5 4 -2 + q - -- + -- - -- + -- - -- + - + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 14]}
In[12]:=	A2Invariant[Knot[8, 14]][q]
Out[12]=	-22 -20 -18 -16 -14 -12 -6 -4 2 4 q - q - q + q - q + q + q - q + -- + q 2 q
In[13]:=	Kauffman[Knot[8, 14]][a, z]
Out[13]=	3 5 7 2 2 2 4 2 6 2 1 + a z + 3 a z + 3 a z + a z - 2 z - a z + 3 a z + a z - 8 2 3 3 3 5 3 7 3 4 2 4 4 4 a z - 3 a z - 6 a z - 8 a z - 5 a z + z - a z - 7 a z - 6 4 8 4 5 3 5 5 5 7 5 2 6 4 a z + a z + 2 a z + 3 a z + 4 a z + 3 a z + 2 a z + 4 6 6 6 3 7 5 7 5 a z + 3 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[8, 14]], Vassiliev[3][Knot[8, 14]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[8, 14]][q, t]
Out[15]=	2 3 1 2 1 2 2 3 2 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 3 3 2 3 t 3 2 ----- + ----- + ---- + ---- + - + q t + q t 7 2 5 2 5 3 q q t q t q t q t